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Vol.1, 2012 ISBN 978-1-59973-186-5 Mathematical Combinatorics (International Book Series) Edited By Linfan MAO The Madis of Chinese Academy of Sciences June, 2012 Aims and Scope: TThe Mathematical Combinatorics (International Book Series) (ISBN 978-1-59978-186-5) is a fully refereed international book series, published in USA quar- terly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, math- ematical combinatorics, non-euclidean geometry and topology and their applications to other sciences. Topics in detail to be covered are: Smarandache multi-spaces with applications to other sciences, such as those of algebraic multi-systems, multi-metric spaces,---, etc.. Smarandache geometries; Differential Geometry; Geometry on manifolds; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Low Dimensional Topology; Differential Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations with Manifold Topology; Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi- natorics to mathematics and theoretical physics; Mathematical theory on gravitational fields; Mathematical theory on parallel universes; Other applications of Smarandache multi-space and combinatorics. Generally, papers on mathematics with its applications not including in above topics are also welcome. It is also available from the below international databases: Serials Group/Editorial Department of EBSCO Publishing 10 Estes St. Ipswich, MA 01938-2106, USA Tel.: (978) 356-6500, Ext. 2262 Fax: (978) 356-9371 http://www.ebsco.com/home/printsubs/priceproj.asp and Gale Directory of Publications and Broadcast Media, Gale, a part of Cengage Learning 27500 Drake Rd. Farmington Hills, MI 48331-3535, USA Tel.: (248) 699-4253, ext. 1326; 1-800-347-GALE Fax: (248) 699-8075 http://www.gale.com Indexing and Reviews: Mathematical Reviews(USA), Zentralblatt fur Mathematik(Germany), Referativnyi Zhurnal (Russia), Mathematika (Russia), Computing Review (USA), Institute for Scientific Information (PA, USA), Library of Congress Subject Headings (USA). Subscription A subscription can be ordered by an email to 7. mathematicalcombinatorics @gmail.com or directly to Linfan Mao The Editor-in-Chief of International Journal of Mathematical Combinatorics Chinese Academy of Mathematics and System Science Beijing, 100190, P.R.China Email: maolinfan@163.com Price: US$48.00 Editorial Board (2nd) Editor-in-Chief Linfan MAO Chinese Academy of Mathematics and System Science, P.R.China and Beijing University of Civil Engineering and Ar- chitecture, P.R.China Email: maolinfan@163.com Editors S.Bhattacharya Deakin University Geelong Campus at Waurn Ponds Australia Email: Sukanto.Bhattacharya@Deakin.edu.au Dinu Bratosin Institute of Solid Mechanics of Romanian Ac- ademy, Bucharest, Romania Junliang Cai Beijing Normal University, P.R.China Email: caijunliang@bnu.edu.cn Yanxun Chang Beijing Jiaotong University, P.R.China Email: yxchang@center.njtu.edu.cn Shaofei Du Capital Normal University, P.R.China Email: dushf@mail.cnu.edu.cn Xiaodong Hu Chinese Academy of Mathematics and System Science, P.R.China Email: xdhu@amss.ac.cn Yuanqiu Huang Hunan Normal University, P.R.China Email: hyqq@public.cs.hn.cn H.Iseri Mansfield University, USA Email: hiseri@mnsfld.edu Xueliang Li Nankai University, P.R.China Email: lxl@nankai.edu.cn Guodong Liu Huizhou University Email: Igd@hzu.edu.cn Ion Patrascu Fratii Buzesti National College Craiova Romania Han Ren East China Normal University, P.R.China Email: hrenQ@math.ecnu.edu.cn Ovidiu-Tlie Sandru Politechnica University of Bucharest Romania. Tudor Sireteanu Institute of Solid Mechanics of Romanian Ac- ademy, Bucharest, Romania. W.B.Vasantha Kandasamy Indian Institute of Technology, India Email: vasantha@iitm.ac.in Luige Vladareanu Institute of Solid Mechanics of Romanian Ac- ademy, Bucharest, Romania Mingyao Xu Peking University, P.R.China Email: xumy@math.pku.edu.cn Guiying Yan Chinese Academy of Mathematics and System Science, P.R.China Email: yanguiying@yahoo.com Y. Zhang Department of Computer Science Georgia State University, Atlanta, USA Only those who dare to fail greatly can ever achieve greatly. By John Kennedy, the 35th President of the United States. Math. Combin. Book Ser. Vol.2(2012), 1-8 Neutrosophic Rings IT Agboola A.A.A., Adeleke E.O. and Akinleye S.A. (Department of Mathematics, University of Agriculture, Abeokuta, Nigeria) E-mail: aaaola2003@yahoo.com, yemi376@yahoo.com, akinleyesa@yahoo.com Abstract: This paper is the continuation of the work started in [12]. The present paper is devoted to the study of ideals of neutrosophic rings. Neutrosophic quotient rings are also studied. Key Words: Neutrosophic ring, neutrosophic ideal, pseudo neutrosophic ideal, neutro- sophic quotient ring. AMS(2010): 03B60, 12E05, 97H40 §1. Introduction The concept of neutrosophic rings was introduced by Vasantha Kandasamy and Florentin Smarandache in [1] where neutrosophic polynomial rings, neutrosophic matrix rings, neutro- sophic direct product rings, neutrosophic integral domains, neutrosophic unique factorization domains, neutrosophic division rings, neutrosophic integral quaternions, neutrosophic rings of real quarternions, neutrosophic group rings and neutrosophic semigroup rings were studied. In [12], Agboola et al further studied neutrosophic rings. The structure of neutrosophic polynomial rings was presented. It was shown that division algorithm is generally not true for neutrosophic polynomial rings and it was also shown that a neutrosophic polynomial ring (RU I) [a] can- not be an Integral Domain even if R is an Integral Domain. Also in [12], it was shown that (RU I) [x] cannot be a unique factorization domain even if R is a unique factorization domain and it was also shown that every non-zero neutrosophic principal ideal in a neutrosophic poly- nomial ring is not a neutrosophic prime ideal. The present paper is however devoted to the study of ideals of neutrosophic rings and neutrosophic quotient rings are also studied. §2. Preliminaries and Results For details about neutrosophy and neutrosophic rings, the reader should see [1] and [12]. Definition 2.1 Let (R,+,-) be any ring. The set (RUI) ={a+bl:a,b€ R} is called a neutrosophic ring generated by R and I under the operations of R, where I is the neutrosophic element and I? = I. 1Received March 14, 2012. Accepted June 2, 2012. 2 Agboola A.A.A., Adeleke E.O. and Akinleye S.A. If (RUD) = (Zn UT) with n < 00, then 0((Z, UI)) = n?. Such a (RU) is said to be a commutative neutrosophic ring with unity if rs = sr for all r,s € (RU J) and 1 € (RUD). Definition 2.2 Let (RU I) be a neutrosophic ring. A proper subset P of (RLJI) is said to be a neutrosophic subring of (RU I) if P=(SUnI), where S is a subring of R and n an integer, P is said to be generated by S and nI under the operations of R. Definition 2.3 Let (RUI) be a neutrosophic ring and let P be a proper subset of (RU 1) which is just a ring. Then P is called a subring. Definition 2.4 Let T be a non-empty set together with two binary operations + and -. T is said to be a pseudo neutrosophic ring if the following conditions hold: (1) T contains elements of the form a+ bl, where a and b are real numbers and b # 0 for at least one value; (2) (T,+) is an abelian group; (3) (L,-) is a semigroup; (4) Va,y,z2€T, e(y+2z) =ayt+ az and (yt z)a =yust zu. Definition 2.5 Let (RU JI) be any neutrosophic ring. A non-empty subset P of (RU I) is said to be a neutrosophic ideal of (RI) if the following conditions hold: (1) P is a neutrosophic subring of (RUD); (2) for everyp € P andr € (RU), rp € P and pr € P. If only rp € P, we call P a left neutrosophic ideal and if only pr € P, we call P a right neutrosophic ideal. When (RU I) is commutative, there is no distinction between rp and pr and therefore P is called a left and right neutrosophic ideal or simply a neutrosophic ideal of (RUD). Definition 2.6 Let (RU JI) be a neutrosophic ring and let P be a pseudo neutrosophic subring of (RU I). P is said to be a pseudo neutrosophic ideal of (RU I) if Vp € P andr € (RUD), rp, pre P. Example 2.7 Let (Z UI) be a neutrosophic ring of integers and let P = (nZ UI) for a positive integer n. Then P is a neutrosophic ideal of (Z U I). y z Example 2.8 Let (RU J) = : 2,y,z © (RUT) ? be the neutrosophic ring of 2 x 2 y 0 matrices and let P = : ay © (RUT) >. Then P is a neutrosophic ideal of (ZU I). Theorem 2.9 Let (Z,UI) be a neutrosophic ring of integers modulo p, where p is a prime number. Then: (1) (Z, UL) has no neutrosophic ideals and (2) (Zp, UL) has only one pseudo neutrosophic ideal of order p. Neutrosophic Rings II 3 Proposition 2.10 Let P, J and Q be neutrosophic ideals (resp. pseudo neutrosophic ideals) of a neutrosophic ring (RUT). Then 1) P+ J is a neutrosophic ideal (resp. pseudo neutrosophic ideal) of (RU I); 2) PJ is a neutrosophic ideal (resp. pseudo neutrosophic ideal) of (RU I); 3) PO J is a neutrosophic ideal (resp. pseudo neutrosophic ideal) of (RU I); 4) PQ) = (PJ)Q; 5) PO +Q) =P + PQ; (1) (2) (3) (4) (5) (6) 6) (J+ Q)P=JP+QP. Proof The proof is the same as in the classical ring. Proposition 2.11 Let (RUT) be a neutrosophic ring and let P be a subset of (RUT) t. Then P is a neutrosophic ideal (resp. pseudo neutrosophic ideal) iff the following conditions hold: (1) P40; (2)a,be P=a-—beP; (3)aE Pre (RUI) Sra,ar € P. Proof the proof is the same as in the classical ring. Proposition 2.12 Let (RUT) be any neutrosophic ring. Then (RUT) and < 0 > are neutro- sophic ideals of (RUT). Proposition 2.13 Let (RUT) be a neutrosophic ring with unity (no unit in (RUT) since I~" does not exist in (RUI)) and let P be a neutrosophic ideal of (RUI). If 1 € P then =(RUI). Proposition 2.14 Let (RUT) be a neutrosophic ring with unity (no unit in (RUT) since I~+ does not exist in (RUI)) and let P be a pseudo neutrosophic ideal of (RUT). If1€ P then P#(RUI). Proof Suppose that P is a pseudo neutrosophic ideal of the neutrosophic ring (R U I) with unity and suppose that 1 € P. Let r be an arbitrary element of (R U I). Then by the definition of P, r.1 =r should be an element of P but since P is not a neutrosophic subring of (RU J), there exist some elements b = x + yI with x,y 4 0 in (RUT) which cannot be found in P. Hence P 4 (RU I). Proposition 2.15 Let (RUT) be a neutrosophic ring and leita =x+ yl be a fixed element of (RUT). Suppose that P= {ra:r€ (RUT)} is a subset of (RUT). (1) Ifz,y £0, then P is a left neutrosophic ideal of (R UI); (2) Ifa =0, then P is a left pseudo neutrosophic ideal of (RUT). Proof (1) is clear. For (2), if « = 0 then each element of P is of the form sI for some s € R. Hence P = {0, sI} which is a left pseudo neutrosophic ideal of (R U J). 4 Agboola A.A.A., Adeleke E.O. and Akinleye S.A. Theorem 2.16 Every ideal of a neutrosophic ring (RUT) is either neutrosophic or pseudo neutrosophic. Proof Suppose that P is any ideal of (R UJ). If P 4 (0) or P# (RU J), then there exists a subring S of R such that for a positive integer n, P=< SUnI >. Letp€ Pandreé (RUI). By definition of P, rp,pr € P and the elements rp and pr are clearly of the form a+ b/ with at least b 4 0. Definition 2.17 Let (RUT) be a neutrosophic ring. (1) If P is a neutrosophic ideal of (RUT) generated by an element r =a+bI € (RUT) with a,b #0, then P is called a neutrosophic principal ideal of (RU I), denoted by (r). (2) If P is a pseudo neutrosophic ideal of (R UTI) generated by an element r = aI € (RUT) with a #0, then P is called a pseudo neutrosophic principal ideal of (RUT), denoted by (r). Proposition 2.18 Let (RU I) be a neutrosophic ring and letr = a+bI € (RUT) with a,b 40. (1) (r) is the smallest neutrosophic ideal of (R UTI) containing r; (2) Every pseudo neutrosophic ideal of (RUT) is contained in some neutrosophic ideal of (RUT). Proposition 2.19 Every pseudo neutrosophic ideal of (Z UTI) is principal. Definition 2.20 Let (RUT) be a neutrosophic ring and let P be a neutrosophic ideal (resp. pseudo neutrosophic ideal) of (RUT). (1) P is said to be maximal if for any neutrosophic ideal (resp. pseudo neutrosophic ideal) J of (RUT) such that P C J we have either J= M or J=(RUI). (2) P is said to be a prime ideal if for any two neutrosophic ideals (resp. pseudo neutro- sophic ideals) J and Q of (RU I) such that JQ C P we have either JC P or QC P. Proposition 2.21 Let (RUT) be a commutative neutrosophic ring with unity and let P be a neutrosophic ideal (resp. pseudo neutrosophic ideal) of (RUT). Then P is prime iff xy © P with « and y in (RUT) implies that either x € P ory€ P. Example 2.22 In (ZU) the neutrosophic ring of integers: (1) (nI) where n is a positive integer is a pseudo netrosophic principal ideal. (2) (Z) is the only maximal pseudo neutrosophic ideal. (3) (0) is the only prime neutrosophic ideal (resp. prime pseudo neutrosophic ideal). Definition 2.23 Let (RUT) be a commutative neutrosophic ring and let ¢ = a+ bI be an element of (RUT) with a,be R. (1) If a,b 4 0 and there exists a positive integer n such that x” = 0 then x is called a strong neutrosophic nilpotent element of (RUT). (2) Ifa=0,b40 and there exists a positive integer n such that x” = 0 then x is called a weak neutrosophic nilpotent element of (RUT). Neutrosophic Rings II 5 (3) If b = 0 and there exists a positive integer n such that x” = 0 then x is called an ordinary nilpotent element of (RUT). Example 2.24 In the neutrosophic ring (Z,U J) of integers modulo 4, 0 and 2 are ordinary nilpotent elements, 2/ is a weak neutrosophic nilpotent element and 2 + 2/ is a strong neutro- sophic nilpotent element. Proposition 2.25 Let (RUT) be a commutative neutrosophic ring. (1) The set of all strong neutrosophic nilpotent elements of (RUT) is not a neutrosophic ideal. (2) The set of all weak neutrosophic nilpotent elements of (RUT) is not a neutrosophic ideal. (3) The set of all nilpotent (ordinary, strong and weak neutrosophic) elements of the com- mutative neutrosophic ring (RU I) is a neutrosophic ideal of (RU I). Definition 2.26 Let (RU I) be a neutrosophic ring and let P be a neutrosophic ideal of (RUT). Let (RUI)/P be a set defined by (RUD) /P={r+P:re(RUD}. If addition and multiplication in (RU I) /P are defined by (r+ P)+(s+P)=(r+s)+P, (r+ P)(s+P)=(rs)+P,r,pe (RU), it can be shown that (RU I) /P is a neutrosophic ring called the neutrosophic quotient ring with P as an additive identity. Definition 2.27 Let (RU I) be a neutrosophic ring and let P be a subset of (RU I). (1) If P is a neutrosophic ideal of (RUI) and (RUT) /P is just a ring, then (RU I) /P is called a false neutrosophic quotient ring; (2) If P is a pseudo neutrosophic ideal of (RUI) and (RUI)/P is a neutrosophic ring, then (RU I) /P is called a pseudo neutrosophic quotient ring; (3) If P is a pseudo neutrosophic ideal of (RUI) and (RUI)/P is just a ring, then (RUT) /P is called a false pseudo neutrosophic quotient ring. Example 2.28 Let < Z,UJ >= {0,1,2,3,4,5, J, 21,37, 47,57,1+/,14+27,14+37,1+47,1+ 61,24+/7,2+21,24+3/7,24+4/7,24+57,34+/,34+ 27,34+31,3+47,3+57,44 7,44 27,44 3/,44 47,44+57,5+7,54+ 27,54 37,5+ 47,545} be a neutrosophic ring of integers modulo 6. (1) If P = {0,2, 7,27, 37,47, 57,2+/,2+27,2+37,2+4]7,2+5/}, then P is a neutrosophic ideal of < Zg UI > but < Zg UI > /P={P,1+P,34+P,4+P,5+4+ P} is just a ring and thus < Z,UI > /P isa false neutrosophic quotient ring. (2) If P = {0, 27,4}, then P is a pseudo neutrosophic ideal of < ZUJI > and the quotient ring 6 Agboola A.A.A., Adeleke E.O. and Akinleye S.A. <Z,UI>/P={P14+P2+P,34+P44+P54+PIJ4+P,14+D4+P(24+)/D4+P,384D+ P,(4+1)+P,(5+J) + P} is a pseudo neutrosophic quotient ring. (3) If P = {0,/,27,37,47,5I}, then P is a pseudo neutrosophic ideal and the quotient ring. < Z.UI > /P={P1+P,2+P,3+P,44+ P,5+ P} is a false pseudo neutrosophic quotient ring. Definition 2.29 Let (RUT) and (SUI) be any two neutrosophic rings. The mapping ¢ : (RUT) > (SUT) is called a neutrosophic ring homomorphism if the following conditions hold: (1) ¢ is a ring homomorphism; (2) eZ) =I. If in addition @ is both 1—1 and onto, then @ is called a neutrosophic isomorphism and we write (RUT) = (SUT). The set {x € (RUT) : 6(x) = 0} is called the kernel of ¢ and is denoted by Kerd. Theorem 2.30 Let 6: (RUI) — (SUI) be a neutrosophic ring homomorphism and let K = Ker@ be the kernel of ¢. Then: (1) K ts always a subring of (RU I); (2) K cannot be a neutrosophic subring of (RU I); (3) K cannot be an ideal of (RU I). Example 2.31 Let (Z UI) be a neutrosophic ring of integers and let P = 5ZUI. It is clear that P is a neutrosophic ideal of (ZU I) and the neutrosophic quotient ring (Z U I) /P is obtained as (ZUD/P = {P1+P,2+P3+P4+P,I4+P,21+P,31+P,4I1+P, (1+D+P,(1+2N+Ph,14+3N+R1+4D+PR,2+D+P, (2421) +P,(24+3ND+P,(2+4)+P,(3+1D+P,(3+21) +P, (G4INEP BEANE RAED P44 2)4 P6443) +P 4a EP, The following can easily be deduced from the example: (1) (ZU TJ) /P is neither a field nor an integral domain. (2) (Z UT) /P and the neutrosophic ring < Z; UI > of integers modulo 5 are structurally the same but then (3) The mapping ¢: (ZUI) — (ZU I) /P defined by ¢(z) = x+ P for all x € (ZU) is not a neutrosophic ring homomorphism and consequently (Z UI) 4 (Z UTI) /P ##< Z5,UI >. These deductions are recorded in the next proposition. Proposition 2.32 Let (Z UI) be a neutrosophic ring of integers and let P = (nZ UTI) where n is a positive integer. Then: (1) (ZUT) /P is a neutrosophic ring; (2) (ZUT) /P is neither a field nor an integral domain even if n is a prime number; (3) (ZU) /P # (ZU). Neutrosophic Rings II 7 Theorem 2.33 If P is a pseudo neutrosophic ideal of the neutrosophic ring (Zp, UTI) of integers modulo n, then (Z, UT) /P & Zp. Proof Let P = {0,/,21,31,--- ,(m—3)I, (n— 2)I, (n—1)J}. It is clear that P is a pseudo neutrosophic ideal of (Z,, UI) and (Z, UI) /P is a false neutrosophic quotient ring given by (Z, UT) /P={P,14+P,2+P,3+P,---,(n—-3)+P,(n—2)+P,(n-1) +P} 2% Zn. Proposition 2.34 Let 6: (RUI) — (SUI) be a neutrosophic ring homomorphism. (1) The set ((RUJ)) = {g(r) :r € (RUD} is a neutrosophic subring of (SUI); (2) o(-r) =—9(r) Vr e (RUT); (3) If 0 is the zero of (RUT), then $(0) is the zero of ¢((RUI)); (4) If P is a neutrosophic ideal (resp. pseudo neutrosophic ideal) of (RU I), then o(P) is a neutrosophic ideal (resp. pseudo neutrosophic ideal) of (SU I); (5) If J is a neutrosophic ideal (resp. pseudo neutrosophic ideal) of (SUT), then ¢~1(J) is a neutrosophic ideal (resp. pseudo neutrosophic ideal) of (RU I); (6) If (RUT) has unity 1 and 6(1) £0 in (SUT), then $(1) is the unity o((RU J)); (7) If (RUT) is commutative, then d((RUI)) is commutative. Proof The proof is the same as in the classical ring. References [1] Vasantha Kandasamy W.B. and Florentin Smarandache, Neutrosophic Rings, Hexis, Phoenix, Arizona, 2006. [2] Vasantha Kandasamy W.B. and Florentin Smarandache, Some Neutrosophic Algebraic Structures and Neutrosophic N-Algebraic Structures, Hexis, Phoenix, Arizona, 2006. [3 Vasantha Kandasamy W.B. and Florentin Smarandache, Basic Neutrosophic Algebraic Structures and their Applications to Fuzzy and Neutrosophic Models, Hexis, Church Rock, 2004. 4) Vasantha Kandasamy W.B., Gaussian polynomial rings, Octogon, vol.5, 58-59, 1997. 5] Vasantha Kandasamy W.B., Inner associative rings, J. of Math. Res. and Expo., vol.18, 217-218, 1998. 6] Vasantha Kandasamy W.B., CN rings, Octogon, vol.9, 343-344, 2001. 7| Vasantha Kandasamy W.B., On locally semi unitary rings, Octogon, vol.9, 260-262, 2001. 8] Vougiouklis Thomas, On rings with zero divisors strong V-groups, Comment Math. Univ. Carolin J., vol.31, 431-433, 1990. 9] Wilson John S., A note on additive subgroups of finite rings, J. Algebra, vol.234, 362-366, 2000. [10] Florentin Smarandache, A Unifying Field in Logics: Neutrosophic Logic, Neutrosophy, Neutrosophic Set, Neutrosophic Probability, (3rd edition), American Research Press, Re- hoboth, 2003. 11 12 13 14 15 16 17 Agboola A.A.A., Adeleke E.O. and Akinleye S.A. Agboola A.A.A. and Akinola L.S., On the Bicoset of a Bivector Space, Int. J. of Math. Comb., vol.4, 1-8, 2009. Agboola A.A.A., Akinola A.D., and Oyebola O.Y., Neutrosophic Rings I, Int. J. of Math. Comb., vol.4, 1-14, 2011. Fraleigh, J.B., A First Course in Abstract Algebra (5th Edition), Addison-Wesley, 1994. Lang S., Algebra, Addison Wesley, 1984. Atiyah M.F. and MacDonald I.G., Introduction to Commutative Algebra, Addison-Wesley, 1969. Herstein I.N., Topics in Algebra, John Wiley and Sons, 1964. Herstein I.N., Topics in Ring Theory, University of Chicago Press, 1969. Math. Combin. Book Ser. Vol.2(2012), 9-23 Non-Solvable Spaces of Linear Equation Systems Linfan Mao (Chinese Academy of Mathematics and System Science, Beijing 100190, P.R.China) E-mail: maolinfan@163.com Abstract: A Smarandache system (X;R) is such a mathematical system that has at least one Smarandachely denied rule in R, i.e., there is a rule in (©; R) that behaves in at least two different ways within the same set ©, i.e., validated and invalided, or only invalided but in multiple distinct ways. For such systems, the linear equation systems without solutions, i.e., non-solvable linear equation systems are the most simple one. We characterize such non- solvable linear equation systems with their homeomorphisms, particularly, the non-solvable linear equation systems with 2 or 3 variables by combinatorics. It is very interesting that every planar graph with each edge a straight segment is homologous to such a non-solvable linear equation with 2 variables. Key Words: Smarandachely denied axiom, Smarandache system, non-solvable linear equa- tions, V-solution, A-solution. AMS(2010): 15A06, 68R10 §1. Introduction Finding the exact solution of equation system is a main but a difficult objective unless the case of linear equations in classical mathematics. Contrary to this fact, what is about the non-solvable case? In fact, such an equation system is nothing but a contradictory system, and characterized only by non-solvable equations for conclusion. But our world is overlap and hybrid. The number of non-solvable equations is more than that of the solvable. The main purpose of this paper is to characterize the behavior of such linear equation systems. Let R™, R™ be Euclidean spaces with dimensional m, n > 1 and 7: R” x R” — R™ be a C*, 1 <k < oo function such that T(Zo, Jo) = 0 for Fo € R”, Jy € R™ and the m x m matrix OT! /Oy' (Zo, Fo) is non-singular, i.e., aet( >) | (Zoo) #0, wherel < i,j <m. Then the implicit function theorem ([1]) implies that there exist opened neighborhoods V Cc R” of Zo, W C R™ of J and a C* function ¢: V — W such that T(z, 6(@)) =0. Thus there always exists solutions for the equation T(Z, (y)) = 0 if T is C’, 1< k < oo. Now let T,,T2,--: ,Im, m> 1 be different C* functions R” x R™ — R™ for an integer k > 1. An 1Received March 6, 2012. Accepted June 5, 2012. 10 Linfan Mao equation system discussed in this paper is with the form following for all integers 1 < ig < m. Denoted by S$® the solutions of equation T;(Z,7) = 0 for integers m m 1<i<m. Then U S$? and () 9° are respectively the V-solutions and A-solutions of equations j= i=l (Eq). By definition, we are easily knowing that the A-solution is nothing but the same as the classical solution. Definition 1.1 The V-solvable, \-solvable and non-solvable spaces of equations (Eq) are re- spectively defined by Us? () 3° and UJ s?- (8°. i=l i=l i=1 i=l Now we construct a finite graph G[Eq] of equations (Eq) following: V(G[Eq]) = {ull Sis m}, E(G[Eq)) = {(v, 09) Such a graph G[Eq] can be also represented by a vertex-edge labeled graph G"[Eq] following: A(Zo, Yo) > T; (Zo, Yo) =0A T; (Zo, Yo) =0,; 1 < 1, J ‘< m}. V(G*[Eq]) = {S2|1 <i < m}, E(G[Eq]) = {(S9, $9) labeled with S°()$9|S9 1 S9 40,1 < i,j < m}. 4? J For example, let S? = {a,b,c}, S? = {c,d,e}, S$ = {a,c,e} and S$? = {d,e, f}. Then its edge-labeled graph G[Eq] is shown in Fig.1 following. {o} 51) 8 {a, c} Non-Solvable Spaces of Linear Equation Systems 11 Notice that U) S$? = LU S?, ie., the non-solvable space is empty only if m = 1 in (Eq). i=1 i= Generally, let (1;R1) (S2;R2), --- ,(2mj;Rm) be mathematical systems, where R; is a rule on 4; for integers 1 <i < m. If for two integers i,j, 1 < i,j <m, 4; A 4; or L; = Uy but Ri # Rj, then they are said to be different, otherwise, identical. Definition 1.2((12]-[13]) A rule in R a mathematical system (U; R) is said to be Smarandachely denied if it behaves in at least two different ways within the same set %, 1.e., validated and invalided, or only invalided but in multiple distinct ways. A Smarandache system (X;R) is a mathematical system which has at least one Smaran- dachely denied rule in R. Thus, such a Smarandache system is a contradictory system. Generally, we know the conception of Smarandache multi-space with its underlying combinatorial structure defined following. Definition 1.3((8]-[10]) Let (21; R1), (H2; Re), +--+, (Um; Rm) bem > 2 mathematical spaces, different two by two. A Smarandache multi-space ¥ is a union U bj with rules R= U Ri on i=1 i=1 ay i.e., the rule Ry on %; for integers 1<1<m, denoted by (E:R). Similarly, the underlying graph of a Smarandache multi-space (E:R) is an edge-labeled graph defined following. Definition 1.4((8]-[10]) Let (5:) be a Smarandache multi-space with © = U d; and R= w=1 U R;. Its underlying graph G [E.R is defined by i=1 V (c [5, R]) = {2n, Ue,-:-, Em}, B(G|5,R]) ={ (2,2,) | 2) 401 S45 sm} with an edge labeling IF: (X;,0;)€ E (c [S. R|) = 17(5,,5,) =o (2 =;) where w is a characteristic on X; (|X; such that U;() Xj is isomorphic to Xp () Xi if and only if @(Xi (25) = w (Ue f) 1) for integers 1 < i,j, k,l <m. We consider the simplest case, i.e., all equations in (Eq) are linear with integers m > n and m,n > 1 in this paper because we are easily know the necessary and sufficient condition of a linear equation system is solvable or not in linear algebra. For terminologies and notations not mentioned here, we follow [2]-[3] for linear algebra, [8] and [10] for graphs and topology. Let AX = (b1,bo,+-+ , bm)? (LEq) be a linear equation system with 12 Linfan Mao Q11 a12 *** Gin Ty a21 G22 *** Gan v2 A= and X = aml aAm2 oe Amn In for integers m, n > 1. Define an augmented matrix At of A by (b1,b2,--- , bm)? following: Q11 a12 *** Gin by At= a21 a22 *** G2n be aml aAm2 ee Amn bm We assume that all equations in (LFq) are non-trivial, i.e., there are no numbers \ such that (441, @i2,°++ , Qin, bs) = A(Qj1, Gj2,*-+ , jn, b;) for any integers 1 < i,7 <m. Such a linear equation system (LEq) is non-solvable if there are no solutions 7;, 1<i<n satisfying (LEq). §2. A Necessary and Sufficient Condition for Non-Solvable Linear Equations The following result on non-solvable linear equations is well-known in linear algebra((2]-[3]). Theorem 2.1 The linear equation system (LEq) is solvable if and only if rank(A) = rank(AT). Thus, the equation system (LEq) is non-solvable if and only if rank(A) 4 rank(AT). We introduce the conception of parallel linear equations following. Definition 2.2 For any integers 1<i,7 <m, i#j, the linear equations Q41 21 + AyQ%Q + +++ Ginn = bi, 05121 + Aj2%2 + +++ AjnTn = b; are called parallel if there exists a constant c such that C= 451/Gi1 = Aj2/ai2 = +++ = Ajn/ Ain F b;/bi. Then we know the following conclusion by Theorem 2.1. Corollary 2.3 For any integers i,j, 1 #7, the linear equation system A412] + AQX2 + +++ AinTn = bi, Qj1 21 + Aj2v2 spite Ajntn = b; is non-solvable if and only if they are parallel. Non-Solvable Spaces of Linear Equation Systems 13 Proof By Theorem 2.1, we know that the linear equations A412] + A272 + +++ AinTn = bi, aj1T1 1 AjQ2XQ aes AjnIn = b; is non-solvable if and only if rankA’ 4 rankB’, where Qt Gi2 *** Ain Gil iQ + Ain Dy A= , B= Qj1 Aj2 *** Ajn Qj1 GAj2 *** Ayn be It is clear that 1 < rankA’ < rankB’ < 2 by the definition of matrixes A’ and B’. Consequently, rankA’ = 1 and rankB’ = 2. Thus the matrix A’, B’ are respectively elementary equivalent to matrixes 1 0 1 0 0 0 0 0 Onl -Y |e 0 0 ie., there exists a constant c such that ¢ = aj1/ai) = aj2/aig = +++ = Ajn/din but c F b;/by. Whence, the linear equations Qi1L1 + Ajg%2 + +++ GinEn = bi, Aj121 1 Aj2X2Q apie AjnIn = b; is parallel by definition. We are easily getting another necessary and sufficient condition for non-solvable linear equations (LEq) by three elementary transformations on a m x (n + 1) matrix At defined following: (1) Multiplying one row of At by a non-zero scalar c; (2) Replacing the ith row of At by row i plus a non-zero scalar c times row j; (3) Interchange of two row of AT. Such a transformation naturally induces a transformation of linear equation system (LEq), denoted by T(LEq). By applying Theorem 2.1, we get a generalization of Corollary 2.3 for non- solvable linear equation system (LEq) following. Theorem 2.4 A linear equation system (LEgq) is non-solvable if and only if there exists a composition T of series elementary transformations on A* with T(AT) the forms following td / / / Ay, AQ + Ay OY / / / / G1 492, ++ Ag Dg T(At) = / / / / aml am2 amn b 14 Linfan Mao and integers 1, 7 with 1 <i,7 <m such that the equations / / / / i121 + AjgL2 + +++ AinTn = b;, f: / / i Gy Ly + Ajg%Q 1+ Ain tn = by are parallel. Proof Notice that the solution of linear equation system following has exactly the same solution with (LEq). If there are indeed integers k and i, j with 1 < k,7,7 <m such that the equations / / / / Ay T1 + Aigo +++ Ginn = bj, / ! / = it A511 + Ajo%2 +++ Ain Tn = b,; are parallel, then the linear equation system (LEq*) is non-solvable. Consequently, the linear equation system (LEq) is also non-solvable. Conversely, if for any integers k andi, j with 1 < k,i,7 < m the equations / / / t ay T1 + ajo%2 + res Aintn = b,, ! / ? ur 5121 + Ajo%2 +++ Ain En = b; are not parallel for any composition T of elementary transformations, then we can finally get a linear equation system 1, + C1 s41%ig4, Heo + C1 nt, = dh Lig + €2,541%1,4, t°°+ + CanX1, = de (LEq™*) ZX], T Cs,st+1Ule44 ste? + Cs.nt1, = ds by applying elementary transformations on (LEq) from the knowledge of linear algebra, which has exactly the same solution with (LEq). But it is clear that (LEgq**) is solvable, i.e., the linear equation system (LEq) is solvable. Contradicts to the assumption. This result naturally determines the combinatorial structure underlying a linear equation system following. Theorem 2.5 A linear equation system (LEgq) is non-solvable if and only if there exists a composition T of series elementary transformations such that G[T(LEq)] # Km, where Ky, is a complete graph of order m. Non-Solvable Spaces of Linear Equation Systems 15 Proof Let T(A™) be / / / / G1 G2 77° Ain 1 / / / / a a te 21 22 2n 2 T(At) = / / / / Ami am2 a Qmn Din If there are integers 1 < i, 7 <_m such that the linear equations / / / / Gj %1 + Ayg®q + +++ Ayn n = b;, / / / / Ay Ly + Aygo + +++ Ain En = 05 are parallel, then there must be S?(\S) = 0, where S?, S¥ are respectively the solutions of linear equations @j;%1 + @jg%2 + +++ @jn%n = 05 and ai,x1 + aijgx2 + +++ a5j,Xn = bi. Whence, there are no edges (S), S$) in G[LEq] by definition. Thus G[LEq] 4 Km. We wish to find conditions for non-solvable linear equation systems (LEq) without elemen- tary transformations. In fact, we are easily determining G[LEq] of a linear equation system (LEq) by Corollary 2.3. Let L; be the ith linear equation. By Corollary 2.3, we divide these equations L;, 1<%i< m into parallel families C1, G2,°°° 6s by the property that all equations in a family @; are parallel and there are no other equations parallel to lines in @; for integers 1 < i < s. Denoted by |@| = ni, 1 <i < s. Then the following conclusion is clear by definition. Theorem 2.6 Let (LEq) be a linear equation system for integers m,n >1. Then G[LEq| > Mas pean: sn s with ny t+n+24+---+n, = m, where ©; is the parallel family with n; = |@;| for integers 1<i<s in (LEq) and (LEq) is non-solvable if s > 2. Proof Notice that equations in a family @; is parallel for an integer 1 < i < m and each of them is not parallel with all equations in U G. Let n; = |@;| for integers 1 <i < s in 1<l<m fi (LEq). By definition, we know G[LEq| > ona sn s with nj +n+2+4+---+ng =m. Notice that the linear equation system (LEq) is solvable only if G[LEq| ~ Km by definition. Thus the linear equation system (LEq) is non-solvable if s > 2. Notice that the conditions in Theorem 2.6 is not sufficient, i.e., if GILEq] ~ Kny ing, .nes we can not claim that (LEq) is non-solvable or not. For example, let (LEq*) and (LEq**) be 16 Linfan Mao two linear equations systems with 1 0 0O 1 0 0 At = 0 1 O At = 0 1 0 1 1 0 2 2 1 -1 0 i222 Then G[LEq*| ~ G[LEq**| ~ K4. Clearly, the linear equation system (LEq*) is solvable with x1 = 0, 22 = 0 but (LEgq**) is non-solvable. We will find necessary and sufficient conditions for linear equation systems with two or three variables just by their combinatorial structures in the following sections. §3. Linear Equation System with 2 Variables Let AX = (b1,b2,--+ bm)? (LEq2) be a linear equation system in 2 variables with Qa11 a12 a21 a22 XY A= and X= x2 Am1 aAm2 for an integer m > 2. Then Theorem 2.4 is refined in the following. Theorem 3.1 A linear equation system (LEq2) is non-solvable if and only if one of the following conditions hold: (1) there are integers 1 <i,j <m such that aj /aj1 = ai2/ajo 4 bi /d;; (2) there are integers 1 < i,j,k < m such that Ail Ai2 ai Oj Qj1 aj2 4 aj1 b; ail Ai2 ayy Di Aki AK api OK Proof The condition (1) is nothing but the conclusion in Corollary 2.3, i.e., the ith equation is parallel to the jth equation. Now if there no such parallel equations in (LEq2), let T be the elementary transformation replacing all other jth equations by the jth equation plus (—aj1/ai1) Non-Solvable Spaces of Linear Equation Systems 17 times the ith equation for integers 1 < 7 < m. We get a transformation T(At) of At following ail Ai2 ain 0; 0 a11 412 ay by ail Ai2 ai by 0 As1 as2 As1 bs +) _ T(A )= ail ai2 b; ’ ail Ai2 ai Oy 0 at1 4t2 at be ail aj2 Qil bi 0 aml aAm2 aml bm, where s = i—1, ¢ =i+1. Applying Corollary 2.3 again, we know that there are integers 1<1,9,k <m such that ail Qi2 ail b; Qj1 ajo ¥ aj1 b; ail Ai2 ayy bi Akl Ar aki br if the linear equation system (LEQ2) is non-solvable. Notice that a linear equation ax; + br2 = c with a 4 0 or b ¥ 0 is a straight line on R?. We get the following result. Theorem 3.2 A liner equation system (LEq2) is non-solvable if and only if one of conditions following hold: (1) there are integers 1 <i,j <m such that aj /aj1 = ai2/ajo 4 bi /d;; G11 12 (2) let #0 and a21 422 by aay ayy by bz az2 az, be 0 _ 7 — t= ’ 2 aii a2 aii 412 a21 a22 a21 422 Then there is an integer 1, 1 <i<m such that aj (21 — x) + aj2(X2 = re) # 0. 18 Linfan Mao Proof If the linear equation system (LE q2) has a solution (x9, x9), then by aay ayy by bz a22 az bg 0 _ Os y= ’ T= ai, a42 Qai1 412 a21 422 a21 422 and a2} + ai2r8 = bi, ie., ai (a1 — x?) + aio(x2 — x9) = 0 for any integers 1 <i < m. Thus, if the linear equation system (L£q2) is non-solvable, there must be integers 1 < i,7 <m such that aj1/aj1 = ai2/aj2 # b;/b;, or there is an integer 1 < i < m such that aji(a1 — 29) + ajo(ao — x8) £0. This completes the proof. For a non-solvable linear equation system (LE q2), there is a naturally induced intersection- free graph I[LEq2| by (LEq2) on the plane R? defined following: V(I[LEq2]) = {(27 ay )lanay + ary = bi, ajay’ + ajay =b;,1< i,j Sm}. E(I[LEq2]) = {(vij, via) |the segament between points (c'!, ct!) and («i!, ci!) in R?}. (where 0 = (ae) oF) fon < 1,9 < m): Such an intersection-free graph is clearly a planar graph with each edge a straight segment since all intersection of edges appear at vertices. For example, let the linear equation system be (LEq2) with Ar = ee Be Oo NO RF Re EF wo wo bw Then its intersection-free graph I[L.Eq2] is shown in Fig.2. v2 wm=l V14 U45 K 45 U24 1 U24 v13 = RWG V13 = V15 = U35 U24 Non-Solvable Spaces of Linear Equation Systems 19 Let H be a planar graph with each edge a straight segment on R?. Its c-line graph Lo(H) is defined by V(Lc(H)) = {straight lines L = eye2---e;,8 > 1 in A}; E(Lo(H)) = {(L1,L2)| if e; and e7 are adjacent in H for Li = ejej:--e7, Lg = efen-2-e2. Loe Sl}. The following result characterizes the combinatorial structure of non-solvable linear equa- tion systems with two variables by intersection-free graphs I[L Fq2]. Theorem 3.3 A linear equation system (LEq2) is non-solvable if and only if G[LEq2] ~ Lo(H)), where H is a planar graph of order |H| > 2 on R? with each edge a straight segment Proof Notice that there is naturally a one to one mapping ¢ : V(G|LEq2]) — V(Lo([LEq2])) determined by ¢(S?) = Si for integers 1 < i < m, where S? and S} denote respectively the solutions of equation a;12%1 +a;2%2 = b; on the plane R? or the union of points between (2}’, x3’) and (xi!, 2!) with ij tj Airy +4;2%y = b; LI a tnt d ab ary + aj2ry = b; and ay! + ant} = b; ani! + anak = by for integers 1 < i, j,1 < m. Now if ($?, S?) € E(G[LEgq2]), then $? (|S? 4 0. Whence, Si) 3} = 9(S?) ($87) = 9(5? (]S?) #0 by definition. Thus (97,57) € Lo(I(LEq2)). By definition, ¢ is an isomorphism between G|LEq2] and Lco(I|LEq2]), a line graph of planar graph I[LEq2| with each edge a straight segment. Conversely, let H be a planar graph with each edge a straight segment on the plane R?. Not loss of generality, we assume that edges €1,2,--- ,e, € E(#) is on a straight line L with equation ap1%1 + Az2%2 = by. Denote all straight lines in H by @. Then H is the intersection-free graph of linear equation system Gr1%1 + ap2% =bp, LE. (LEq2*) Thus, G|LEq2*| ~ H. This completes the proof. Similarly, we can also consider the liner equation system (LEq2) with condition on x1 or x2 such as AX = (bi, be, +++ ,bm)™ (L~ Eq2) 20 Linfan Mao with Q11 a12 a21 a22 Ty A = 5 xX — XQ Am1 Am2 and x; > x° for a real number x° and an integer m > 2. In geometry, each of there equation is a ray on the plane R?, seeing also references [5]-[6]. Then the following conclusion can be obtained like with Theorems 3.2 and 3.3. Theorem 3.4 A linear equation system (L~ Eq2) is non-solvable if and only if G[LEq2] ~ Lo(H)), where H is a planar graph of order |H| > 2 on R? with each edge a straight segment. §4. Linear Equation Systems with 3 Variables Let AX = (bi, ba,+++ 5m)” (LEq3) be a linear equation system in 3 variables with Q11 a12 a13 a21 a22 a23 A= and X= v2 X3 Gm1 Am2 m3 for an integer m > 3. Then Theorem 2.4 is refined in the following. Theorem 4.1 A linear equation system (LEq3) is non-solvable if and only if one of the following conditions hold: (1) there are integers 1 < 4,9 < m such that a1 /G51 = a2 /Q;2 = 043/053 # b;/b;; (2) if (ai, ai2, ai3) and (aj1, aj2,4;3) are independent, then there are numbers , 1 and an integer 1, 1<1<m such that (ai1, d12, @13) = A(@i1, Gi2, G3) + W(j1, 472, 23) but b # Ab; + pub; ; (3) if (a1, Qi2, Gi3), (@j1, 472, 43) and (x1, @k2,4n3) are independent, then there are num- bers A, p,V and an integer 1, 1<1<m such that (an, ai2, a3) = A(ai1, ai2, ai3) = play, Qj2, aj3) or V (ae, Qk, aK3) but by x dd; + pub; + Vbp. Non-Solvable Spaces of Linear Equation Systems 21 Proof By Theorem 2.1, the linear equation system (LFq3) is non-solvable if and only if 1 < rankA #4 rankAt < 4. Thus the non-solvable possibilities of (Z.Eq3) are respectively rankA = 1, 2 < rankAt < 4, rankA = 2, 3 < rankAt < 4 and rankA = 3, rankAt = 4. We discuss each of these cases following. Case 1 rankA = 1 but 2 < rankAt < 4 In this case, all row vectors in A are dependent. Thus there exists a number 4 such that Ar => a1 / O51 = ai2/ G52 = 43/053 but aN x b;/;. Case 2 rankA = 2, 3 < rankAt <4 In this case, there are two independent row vectors. Without loss of generality, let (@i1, G2, @i3) and (a;1, a2, 4;3) be such row vectors. Then there must be an integer 1, 1 <1 <m such that the [th row can not be the linear combination of the ith row and jth row. Whence, there are numbers 4, js such that (ai1, d12, @13) = A( Gir, Gi2, G3) + W(G;1, 472, a3) Case 3 rankA = 3, rankAt = 4 In this case, there are three independent row vectors. Without loss of generality, let (@i1, G2, @i3), (Aj1, 4;2,4;3) and (ax1,@%2, 4x3) be such row vectors. Then there must be an integer 1, 1 < 1 < m such that the [th row can not be the linear combination of the ith row, jth row and kth row. Thus there are numbers 4, yu, v such that (ai1, d12, @13) = A(@i1, Gi2, G3) + W(Aj1, 472, 473) + V(Ak1, On2, Ok3) but b; A Abj + wb; + vby. Combining the discussion of Case 1-Case 3, the proof is complete. Notice that the linear equation system (LFq3) can be transformed to the following (L Fq3*) by elementary transformation, i.e., each jth row plus —a;3/aj3 times the ith row in (LEq3) for an integer i, 1 <i <m with aj 4 0, / / / 1 \T * A'X = (b),05,--+ 01) (LEgq3*) with / / / Q11 a2 0 by / / / Ait = | i= %-1)2 0 Oy a >) ail aj2 ai3 bi / / / Gita. %\Q+1)2 0 O44 / / / amt ama 0 Une where ay = 4j1 — 473041 /a13, Ajo = G2 — 4724;2/ai3 and bi = b; — aj3b;/ai3 fro integers 1<j<™m. Applying Theorem 3.3, we get the a combinatorial characterizing on non-solvable linear systems (LFq3) following. 22 Linfan Mao Theorem 4.2 A linear equation system (LEq3) is non-solvable if and only if G[LEq3] # Km or G[LEg3*| ~ u+ Lo(H), where H denotes a planar graph with order |H| > 2, sizem—1 and each edge a straight segment, u+ G the join of vertex u with G. Proof By Theorem 2.4, the linear equation system (L£q3) is non-solvable if and only if G|LEq3] # Ky, or the linear equation system (LFq3*) is non-solvable, which implies that the linear equation subsystem following BX" = (b1,-°: 04, 0;4a- + Bee (LEq2*) with ay a> B= and X’= (a1, a2)" / / Gi-1)1 %i-1)2 al a’, (41)1 %i+1)2 / / mi Ame a is non-solvable. Applying Theorem 3.3, we know that the linear equation subsystem (LE q2*) is non-solvable if and only if G[LEq2*] ~ Lc(H)), where H is a planar graph H of size m— 1 with each edge a straight segment. Thus the linear equation system (Lq3*) is non-solvable if and only if G[LEq3*] ~ u+ Lc(#). §5. Linear Homeomorphisms Equations A homeomorphism on R” is a continuous 1 — 1 mapping h : R” > R” such that its inverse h7! is also continuous for an integer n > 1. There are indeed many such homeomorphisms on R”. For example, the linear transformations T on R”. A linear homeomorphisms equation system is such an equation system AX = (b1,b2,+++ , bm)” (L’ Eq) with X = (h(x1), h(x2),--- ,h(an))", where h is a homeomorphism and a1 412 Gin Ae a21 422 Gan die Cea fee's Mies for integers m, n > 1. Notice that the linear homeomorphism equation system ayth(x1) + ajgh(a2) +--+ dinh(an) = bi, aj1h(a1) mi @52(£2) abirais ajnh(an) = b; is solvable if and only if the linear equation system Qji1L1 + AQ%2 + +++ AinTn = bi, 5121 + Aj2%Q +++ AjnTn = b; Non-Solvable Spaces of Linear Equation Systems 23 is solvable. Similarly, two linear homeomorphism equations are said parallel if they are non- solvable. Applying Theorems 2.6, 3.3,4.2, we know the following result for linear homeomor- phism equation systems (L’ Eq). Theorem 5.1 Let (L’Eq) be a linear homeomorphism equation system for integers m,n > 1. Then (1) G[LEq) © Kny.no-n, with ny +n+2+---+ns =m, where GP is the parallel family with n; = |6)| for integers 1 <i <8 in (L"Egq) and (L" Eq) is non-solvable if s > 2; (2) Ifn = 2, (L"Eq) is non-solvable if and only if G[L"Eq ~ Lc(H)), where H is a planar graph of order |H| > 2 on R? with each edge a homeomorphism of straight segment, and if n = 3, (L"Egq) is non-solvable if and only if G[L’Eq| #4 Km or G[LEq3*] ~ ut+ Lo(H), where H denotes a planar graph with order |H| > 2, size m—1 and each edge a homeomorphism of straight segment. 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Vol.2(2012), 24-31 Roman Domination in Complementary Prism Graphs B.Chaluvaraju and V.Chaitra 1(Department of Mathematics, Bangalore University, Central College Campus, Bangalore -560 001, India) E-mail: bchaluvaraju@gmail.com, chaitrashok@gmail.com Abstract: A Roman domination function on a complementary prism graph GG® is a function f : VUV* — {0,1,2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number yr(GG*) of a graph G = (V, £) is the minimum of SS. evuved (a) over such functions, where the complementary prism GG*° of G is graph obtained from disjoint union of G and its complement G° by adding edges of a perfect matching between corresponding vertices of G and G°. In this paper, we have investigated few properties of yr(GG*) and its relation with other parameters are obtained. Key Words: Graph, domination number, Roman domination number, Smarandachely Roman s-domination function, complementary prism, Roman domination of complementary prism. AMS(2010): 05C69, 05070 §1. Introduction In this paper, G is a simple graph with vertex set V(G) and edge set E(G). Let n = |V| and m = |E| denote the number of vertices and edges of a graph G, respectively. For any vertex v of G, let N(v) and N|v] denote its open and closed neighborhoods respectively. ag(G)(ai1(G)), is the minimum number of vertices (edges) in a vertex (edge) cover of G. G9(G)(Gi(G)), is the minimum number of vertices (edges) in a maximal independent set of vertex (edge) of G. Let deg(v) be the degree of vertex v in G. Then A(G) and 6(G) be maximum and minimum degree of G, respectively. If M is a matching in a graph G with the property that every vertex of G is incident with an edge of M, then M is a perfect matching in G. The complement G° of a graph G is the graph having the same set of vertices as G denoted by V° and in which two vertices are adjacent, if and only if they are not adjacent in G. Refer to [5] for additional graph theory terminology. A dominating set D C V for a graph G is such that each v € V is either in D or adjacent to a vertex of D. The domination number 7(G) is the minimum cardinality of a dominating set of G. Further, a dominating set D is a minimal dominating set of G, if and only if for each vertex v € D, D—v is not a dominating set of G. For complete review on theory of domination 1Received April 8, 2012. Accepted June 8, 2012. Roman Domination in Complementary Prism Graphs 25 and its related parameters, we refer [1], [6]and [7]. For a graph G = (V, £), let f : V — {0,1,2} and let (Vo, Vi, V2) be the ordered partition of V induced by f, where V; = {v € V/f(v) =i} and |V;| = n; for 7 = 0,1,2. There exist 1-1 correspondence between the functions f : V > {0,1,2} and the ordered partitions (Vo, Vi, V2) of V. Thus we write f = (Vo, Vi, V2). A function f = (Vo,Vi, V2) is a Roman dominating function (RDF) if V2 > Vo, where > signifies that the set V2 dominates the set Vo. The weight of a Roman dominating function is the value f(V) = \O,-y f(v) = 2|V2| + |Vi]. Roman dominating number yr(G), equals the minimum weight of an RDF of G, we say that a function f = (Vo,Vi, V2) is a yr-function if it is an RDF and f(V) = yr(G). Generally, let J Cc {0,1,2,---,n}. A Smarandachely Roman s-dominating function for an integer s, 2 << s <n ona graph G = (V,E) is a function f:V — {0,1,2,---,n} satisfying the condition that |f(w) — f(v)| > s for each edge wv € E with f(u) or f(v) € I. Particularly, if we choose n = s = 2 and I = {0}, such a Smarandachely Roman s-dominating function is nothing but the Roman domination function. For more details on Roman dominations and its related parameters we refer [3]-[4] and [9]-[11]. In [8], Haynes etal., introduced the concept of domination and total domination in com- plementary prisms. Analogously, we initiate the Roman domination in complementary prism as follows: A Roman domination function on a complementary prism graph GG* is a function f : VUV* = {0,1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number yr(GG°) of a graph G = (V, £) is the minimum of }).eyyye f(#) over such functions, where the complementary prism GG* of G is graph obtained from disjoint union of G and its complement G° by adding edges of a perfect matching between corresponding vertices of G and G°. §2. Results We begin by making a couple of observations. Observation 2.1 For any graph G with order n and size m, m(GG*) = n(n + 1)/2. Observation 2.2 For any graph G, (i) 61(GG°) =n. (ii) a1(GG*) + Bi(GG*) = 2n. Proof Let G be a graph and GG* be its complementary prism graph with perfect matching M. If one to one correspondence between vertices of a graph G and its complement G° in GG*, then GG* has even order and M is a 1-regular spanning sub graph of GG‘°, thus (i) follows and due to the fact of ai(G) + 31(G) = n,(ii) follows. 26 B.Chaluvaraju and V.Chaitra Observation 2.3 For any graph G, V(GG*) =n if and only if G or G° is totally disconnected graph. Proof Let there be n vertices of degree 1 in GG*. Let D be a dominating set of GG* and v be a vertex of G of degree n —1, v € D. In GG*, v dominates n vertices and remaining n — 1 vertices are pendent vertices which has to dominate itself. Hence y(GG°) = n. Conversely, if 7(GG°) =n, then there are n vertices in minimal dominating set D. Theorem 2.1 For any graph G, yr(GG*) = a1(GG*) + B1(GG*) if and only if G being an isolated vertex. Proof If G is an isolated vertex, then GG‘ is Ky and yr(GG*) = 2,a,(GG*) = 1 and Gi(GG*) = 1. Conversely, if yr(GG°) = ai(GG*) + 61(GG*). By above observation, then we have yr(GG*) = 2|V2| + |Vil. Thus we consider the following cases: Case 1 If V2 = ¢,|Vi| = 2, then Vo = ¢ and GG* & Ky. Case 2 If |V2| =1,|Vi| = ¢, then GG* is a complete graph. Hence the result follows. Theorem 2.2 Let G and G° be two complete graphs then GG® is also complete if and only if G= kK. Proof If G = k, then G° = ky, and GG° = K2 which is a complete graph. Conversely, if GG* is complete graph then any vertex v of G is adjacent to n — 1 vertices of G and n vertices of G°. According to definition of complementary prism this is not possible for graph other than Ky. Theorem 2.3 For any graph G, V(GG*) < yr(GG*) < 27(GG*). Further,the upper bound is attained if Vi(GG*) = ¢. Proof Let f = (Vo, Vi, V2) be yr-function. If V2 > Vo and (Vi U V2) dominates GG*, then (GG) < |Vi U Val = |Vi| + 2|Vo| = yr(GG*). Thus the result follows. Let f = (Vo,Vi, V2) be an RDF of GG*° with |D| = 7(GG*). Let V2 = D,V, = ¢ and VY =V-—D. Since f is an RDF and yr(GG*) denotes minimum weight of f(V). It follows yR(GG*) < f(V) = |Vi| + 2|Va| = 2|S| = 27(GG*). Hence the upper bound follows. For graph GG*, let v be vertex not in Vj, implies that either v € V2 or v € Yo. If vu € V2 then v € D, yrR(GGS) = 2|Vo| + |Vi| = 2|D| = 27(GG*). If vu € Y then N(v) C Va or N(v) C Vo as v does not belong to V;. Hence the result. Roman Domination in Complementary Prism Graphs 27 Theorem 2.4 For any graph G, 2 < yr(GG*) < (n+ 1). Further, the lower bound is attained if and only if G = Ky, and the upper bound is attained if G or G® is totally disconnected graph. Proof Let G be a graph with n > 1. If f = {Vo,Vi, Vo} be a RDF of GG*, then yr(GG°) > 2. Thus the lower bound follows. Upper bound is proved by using mathematical induction on number of vertices of G. For n=1, GG° = Ko, 7rR(GG°) =n+1. For n = 2, GG° = Py, yr(GG*) = n+1. Assume the result to be true for some graph H with n — 1 vertices, yr(HH‘°) < n. Let G be a graph obtained by adding a vertex v to H. If v is adjacent to a vertex w in H which belongs to V2, then v € Vo, yrR(GG°) =n < n-+1. If v is adjacent to a vertex either in Vo or Vi, then yr(GG°) =n-+1. If v is adjacent to all vertices of H then yr(GG°) <n <n+1. Hence upper bound follows for any number of vertices of G. Now, we prove the second part. If G © Ky, then yr(GG*°) = 2. On the other hand, if yrR(GG°) = 2 = 2|V2| + |Vi| then we have following cases: Case 1 If |V2| = 1,|Vil = 0, then there exist a vertex v € V(GG°) such that degree of v = (n— 1), thus one and only graph with this property is GGS & Ka. Hence G = Ky. Case 2 If |V2| = 0,|Vi| = 2, then there are only two vertices in the GG° which are connected by an edge. Hence the result. If G is totally disconnected then G‘° is a complete graph. Any vertex v° in G° dominates n vertices in GG°. Remaining n — 1 vertices of GG® are in V,. Hence yr(GG*) =n +1. Proposition 2.1([3]) For any path P, and cycle Cy, with n > 3 vertices, yR(Pn) = Ya(Cn) = [2n/3], where [a] is the smallest integer not less than x. Theorem 2.5 For any graph G, (i) of G=P, with n > 3 vertices, then YR(GG*) = 4+ [2(n — 3)/3]; (it) if G=C, with n> 4 vertices, then yR(GG°) = 4+4 [2(n — 2)/3]. Proof (i) Let G = P, be a path with with n > 3 vertices. Then we have the following cases: Case 1 Let f = (Vo,Vi, V2) be an RDF and a pendent vertex v is adjacent to a vertex wu in G. The vertex v° is not adjacent to a vertex u° in V°. But the vertex of v° in V° is adjacent 28 B.Chaluvaraju and V.Chaitra to n vertices of GG°. Let v° € V2 and N(v°) C Vo. There are n vertices left and u° € N[u] but {N(u°) — u} C Vo. Hence u € Vo, N(u) C Vo. There are (n — 3) vertices left, whose induced subgraph H forms a path with yr(H) = [2(n—3)/3], this implies that yr(G) = 4+[2(n—3)/3]. Case 2 If v is not a pendent vertex, let it be adjacent to vertices u and w in G. Repeating same procedure as above case , yr(GG°) = 6 + [2(n — 3)/3], which is a contradiction to fact of RDF. (it) Let G=C,, be acycle with n > 4 vertices. Let f = (Vo, Vi, V2) be an RDF and w be a vertex adjacent to vertex u and v in G, and w° is not adjacent to u° and v° in V°. But w° is adjacent to (n — 2) vertices of GG°. Let w° € V2 and N(w°) C Vo. There are (n + 1)-vertices left with u° or v° € V2. With out loss of generality, let uc € V2, N(u°) C Vo. There are (n — 2) vertices left, whose induced subgraph H forms a path with yr(H) = [2(n — 2)/3] and V2 = {w, u°}, this implies that yr(G) = 4+ [2(n — 2)/3]. Theorem 2.6 For any graph G, maz{yr(G),yR(G*)} < yrR(GG*) < (yR(G) + yR(G*)). Further, the upper bound is attained if and only if the graph G is isomorphic with Ky. Proof Let G be a graph and let f : V — {0,1,2} and f = (Vo, Vi, V2) be RDF. Since GG* has 2n vertices when G has n vertices, hence max{yr(G), yr(G°)} < yr(GG*) follows. For any graph G with n > 1 vertices. By Theorem 2.4, we have yr(GG°) < (n+ 1) and (yr (G) + yrR(G*)) < (n+ 2) = (n+ 1) +1. Hence the upper bound follows. Let G = ky. Then GG* = Ko, thus the upper bound is attained. Conversely, suppose G & ky. Let u and v be two adjacent vertices in G and u is adjacent to v and u° in GG*. The set {u,v°} is a dominating set out of which u € V2,u° € Vi. yr(G) = 2, yrR(G°) = 0 and yr(GG°) = 3 which is a contradiction. Hence no two vertices are adjacent in G. Theorem 2.7 If degree of every vertex of a graph G is one less than number of vertices of G, then yr(GG*) = 7(GG*) +1. Proof Let f = (Vo,Vi, V2) be an RDF and let v be a vertex of G of degree n — 1. In GG°,v is adjacent to n vertices. If D is a minimum dominating set of GG*° then v € D, v € V2 also N(v) C Vo. Remaining n — 1 belongs to V; and D. |D| = 7(GG°) = n and yrR(GG°) =n+1=7(GG*) +1. Theorem 2.8 For any graph G with n > 1 vertices, yR(GG*) < [2n — (A(GG*) + 1)]. Further, the bound is attained if G is a complete graph. Proof Let G be any graph with n > 1 vertices. Then GG has 2n- vertices. Let f = (Vo, Vi, V2) be an RDF and v be any vertex of GG such that deg(v) = A(GG*). Then v Roman Domination in Complementary Prism Graphs 29 dominates A(GG°) + 1 vertices. Let v € V2 andN(v) C Vo. There are (2n — (A(GG*) + 1) vertices left in GG*, which belongs to one of Vo, Vi or V2. If all these vertices € Vi, then YR(GG*) = 2|Va| + [Vil = 2 + (2n — A(GG*) + 1) = 2n — A(GG*) + 1. Hence lower bound is attained when G = K,,, where v is a vertex of G. If not all remaining vertices belong to Vi, then there may be vertices belonging to V2 and which implies there neighbors belong to Vo. Hence the result follows. Theorem 2.9 For any graph G, YR(GG*)° < yR(GG*). Further, the bound is attained for one of the following conditions: (i) GG° = (GG); (it) GG®* is a complete graph. Proof Let G be a graph, GG* be its complementary graph and (GG‘°)° be complement of complementary prism. According to definition of GG there should be one to one matching between vertices of G and G°, where as in (GG°)° there will be one to (n—1) matching between vertices of G and G° implies that adjacency of vertices will be more in (GG°)°. Hence the result. If GG° = (GG°)°, domination and Roman domination of these two graphs are same. The only complete graph GG® can be is Ko. (GG*)° will be two isolated vertices, yp(GG*°) = 2 and yr(GG°)° = 2. Hence bound is attained. To prove our next results, we make use of following definitions: A rooted tree is a tree with a countable number of vertices, in which a particular vertex is distinguished from the others and called the root. In a rooted tree, the parent of a vertex is the vertex connected to it on the path to the root; every vertex except the root has a unique parent. A child of a vertex v is a vertex of which v is the parent. A leaf is a vertex without children. A graph with exactly one induced cycle is called unicyclic. Theorem 2.10 For any rooted tree T, yR(TT*) = 2|S2| + [Si], where 5S; C Vi and Sz C Vo. Proof Let T be a rooted tree and f = (Vo, Vi, V2) be RDF of a complementary prism TT*. We label all parent vertices of T as P,, Po, ....P, where P; is root of a tree T’. Let S, be set of all parent vertices of TJ’, S; be set of all leaf vertices of T and v € S; be a vertex farthest from Py. The vertex v° is adjacent to (n — 1)—vertices in TT. Let v° € Sz, and N(v°) C Vo. Let Py be parent vertex of vu € T. For i=1 to k if P; is not assigned weight then P; € S: and N(P;) C Vo. If P; is assigned weight and check its leaf vertices in T, then we consider the following cases: Case 1 If P; has at least 2 leaf vertices, then P; € Sz and N(P;) C VW. 30 B.Chaluvaraju and V.Chaitra Case 2 If P; has at most 1 leaf vertex, then all such leaf vertices belong to S,. Thus yr(GG*) = 2|.S2| + |.S1| follows. Theorem 2.11 Let G° be a complement of a graph G. Then the complementary prism GG° is (i) isomorphic with a tree T if and only if G or G° has at most two vertices. (ii) (n+ 1)/2-regular graph if and only if G is (n — 1)/2-regular. (itt) unicyclic graph if and only if G has exactly 3 vertices. Proof (i) Suppose GG® is a tree T with the graph G having minimum three vertices. Then we have the following cases: Case 1 Let u, v and w be vertices of G with v is adjacent to both u and w. In GG*°, u® is connected to u and w* also v° is connected to v. Hence there is a closed path u-v-w-w*-u°-u, which is a contradicting to our assumption. Case 2 If vertices u, v and w are totally disconnected in G, then G° is a complete graph. Since every complete graph G with n > 3 has cycle. Hence GG* is not a tree. Case 3 If u and v are adjacent but which is not adjacent to w in G, then in GG* there is a closed path u-u°-w%-v°-v°-u, again which is a contradicting to assumption. On the other hand, if G has one vertex, then GG*° = Kz and if G have two vertices, then GG° = P,. In both the cases GG‘* is a tree. (it) Let G be r-regular graph, where r = (n — 1)/2, then G° is nm —r —1 regular. In GG, degree of every vertex in G is r+ 1 = (n+ 1)/2 and degree of every vertex in G° isn —r = (n+ 1)/2, which implies GG° is (n + 1)/2-regular. Conversely, suppose GG° is s = (n+1)/2-regular. Let EF be set of all edges making perfect match between G and G°. In GG — E, G is s—1-regular and G*° is (n—s—1)-regular. Hence the graph G is (n—1)/2-regular. (iti) If GG has at most two vertices, then from (i), GG® is a tree. Minimum vertices required for a graph to be unicyclic is 3. Because of perfect matching in complementary prism and G and G* are connected if there are more than 3 vertices there will be more than 1 cycle. Acknowledgement Thanks are due to Prof. N. D Soner for his help and valuable suggestions in the preparation of this paper. References [1] B.D.Acharya, H.B.Walikar and E.Sampathkumar, Recent developments in the theory of domination in graphs, MRI Lecture Notes in Math., 1 (1979), Mehta Research Institute, Alahabad. Roman Domination in Complementary Prism Graphs 31 B.Chaluvaraju and V.Chaitra, Roman domination in odd and even graph, South East Asian Journal of Mathematics and Mathematical Science (to appear). E.J.Cockayne, P.A.Dreyer Jr, S.M.Hedetniemi and $.T.Hedetniemi, Roman domination in graphs, Discrete Mathematics, 278 (2004) 11-24. O.Favaron, H.Karamic, R.Khoeilar and $.M.Sheikholeslami, Note on the Roman domina- tion number of a graph, Discrete Mathematics, 309 (2009) 3447-3451. F.Harary, Graph theory, Addison-Wesley, Reading Mass (1969). T.W.Haynes, S.T.Hedetniemi and P.J.Slater, Fundamentals of domination in graphs, Mar- cel Dekker, Inc., New York (1998). T.W.Haynes, S.T.Hedetniemi and P.J.Slater, Domination in graphs: Advanced topics, Mar- cel Dekker, Inc., New York (1998). T.W.Haynes, M.A.Henning and L.C. van der Merwe, Domination and total domination in complementary prisms, Journal of Combinatorial Optimization,18 (1)(2009) 23-37. Nader Jafari Rad, Lutz Volkmann, Roman domination perfect graphs, An.st. Univ ovidius constanta., 19(3)(2011)167-174. L.Stewart, Defend the Roman Empire, Sci. Amer., 281(6)(1999)136-139. N.D.Soner, B.Chaluvaraju and J.P.Srivatsava, Roman edge domination in graphs, Proc. Nat. Acad. Sci. India. Sect. A, Vol.79 (2009) 45-50. Math. Combin. Book Ser. Vol.2(2012), 82-88 Enumeration of Rooted Nearly 2-Regualr Simple Planar Maps Shude Long Department of Mathematics, Chongqing University of Arts and Sciences, Chongqing 402160, P.R.China Junliang Cai School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P.R.China E-mail: longshude@163.com; caijunliang@bnu.edu.cn Abstract: This paper discusses the enumeration of rooted nearly 2-regular simple planar maps and presents some formulae for such maps with the valency of the root-face, the numbers of nonrooted vertices and inner faces as three parameters. Key Words: Smarandachely map, simple map, nearly 2-regular map, enumerating func- tion, functional equation, Lagrangian inversion, Lagrangian inversion. AMS(2010): 05C45, 05C30 §1. Introduction Let S be a surface. For an integer k > 0, a Smarandachely k-map on S is such a pseudo-map on S just with & faces that not being 2-cell. If k = 0, such a Smarandachely map is called map. In the field of enumerating planar maps, many functional equations for a variety of sets of planar maps have been found and some solutions of the equations are obtained. Some nice skills are applied in this area and they have set up the foundation of enumerative theory [2], [5], [6] and [9-13]. But the discussion on enumerating function of simple planar maps is very few. All the results obtained so far are almost concentrated in general simple planar maps [3], [4], [7] and [8]. In 1997, Cai [1] investigated for the first time the enumeration of simple Eulerian planar maps with the valency of root-vertex, the number of inner edges and the valency of root-face as parameters and a functional equation satisfied by its enumerating function was obtained, but it is very complicated and its solution has not been found up to now. In this paper we treat the enumeration of rooted nearly 2-regular simple planar maps with the valency of the root-face, the numbers of nonrooted vertices and inner faces as three parameters. Several explicit expressions of its enumerating functions are obtained and one of them is summation-free. Now, we define some basic concepts and terms. In general, rooting a map means distin- guishing one edge on the boundary of the outer face as the root-edge, and one end of that edge as the root-vertex. In diagrams we usually represent the root-edge as an edge with an arrow on 1Supported by the NNSFC under Grant No. 10271017 and Chongqing Municipal Education Commission under Grant No. KJ101204. 2Received February 24, 2012. Accepted June 10, 2012. Enumeration of Rooted Nearly 2-Regualr Simple Planar Maps 33 the outer face, the arrow being drawn from the root-vertex to the other end. So the outer face is also called the root-face. A planar map with a rooting is said to be a rooted planar map. We say that two rooted planar maps are combinatorially equivalent or up to root-preserving iso- morphism if they are related by 1-1 correspondence of their elements, which maps vertices onto vertices, edges onto edges and faces onto faces, which preserves incidence relations and which preserves the root-vertex, root-edge and root-face. Otherwise, combinatorially inequivalent or nonisomorphic here. A nearly 2-regular map is a rooted map such that all vertices probably except the root- vertex are of valency 2. A map is said to be simple, if there is neither loop nor parallel edge. For a set of some maps .@, the enumerating function discussed in this paper is defined as faa = So) eer geo, (1) Meu where [(M),p(M) and q(M) are the root-face valency, the number of nonrooted vertices and the number of inner faces of M/, respectively. Furthermore, we introduce some other enumerating functions for .@ as follows: iaey) = ag, MEA hdd2= yea, MEA Hay) = oy", (2) MEA where 1(M), p(M) and q(M) are the same in (1) and n(M) is the number of edges of M, that is g.a(@,y) =fxu(@,y,Y), hay, 2) = fay, ca Hay) =9.a(1.y) =hayy) = fay, y)- (3) For the power series f(z), f(x,y) and f(x,y, z), we employ the following notations: OP f(a), OP) F(e,y) and a? f(x,y, z) to represent the coefficients of z” in f(a), xy? in f(x,y) and xy? z? in f(a, y, z), respectively. Terminologies and notations not explained here can be found in [11]. §2. Functional Equations In this section we will set up the functional equations satisfied by the enumerating functions for rooted nearly 2-regular simple planar maps. Let & be the set of all rooted nearly 2-regular simple planar maps with convention that the vertex map V is in & for convenience. From the definition of a nearly 2-regular simple map, for any M € &—%, each edge of M is contained in only one circuit. The circuit containing the root-edge is called the root circuit of M, and denoted by C(M). 34 Shude Long and Junliang Cai It is clear that the length of the root circuit is no more than the root-face valency, and E=H+U&, (4) i>3 where & ={M|M € @,the length of C(M) is i} (5) and & is only consist of the vertex map V. It is easy to see that the enumerating function of é is fe(2,y,2) = 1. (6) For any M € 6;(i > 3), the root circuit divides M — C(M) into two domains, the inner domain and outer domain. The submap of M in the outer domain is a general map in &, while the submap of M in the inner domain does not contribute the valency of the root-face of M. Therefore, the enumerating function of &; is fe(a,y,2) = ay 2hf, (7) where h = he(y,z) = fe(1,y, z). Theorem 2.1 The enumerating function f = fe(x,y,z) satisfies the following equation: = xy? zh a: r= 1-7 ®) where h = he(y, z) = fey, @). Proof By (4), (6) and (7), we have f=lt+ . gy’ tzhf i>3 = vy? zhf - l—ay’ which is equivalent to the theorem. Let y = z in (8). Then we have Corollary 2.1 The enumerating function g = ge(x,y) satisfies the following equation: where H = He(y) = ge(1,y). Let x = 1 in (8). Then we obtain Corollary 2.2 The enumerating function h = he(y, z) satisfies the following equation: y’zh? —(1—y)h-y+1=0. (10) Enumeration of Rooted Nearly 2-Regualr Simple Planar Maps 35 Further, let y = z in (10). Then we have Corollary 2.3 The enumerating function H = Hg(y) satisfies the following equation: yH? —-(1-y)H —y+1=0. (11) §3. Enumeration In this section we will find the explicit formulae for enumerating functions f = fe(x,y,z),g = ge(x,y),h = he(y,z) and H = He(y) by using Lagrangian inversion. By (10) we have Let yz = (1 — On). (13) h= Ta (14) By (13) and (14), we have the following parametric expression of h = he(y, z): 0 Y=Typ gp yea nll — On), h=- : i (15) and from which we get lL = Aen) 1-2 seer Theorem 3.1 The enumerating function h = he(y,z) has the following explicit expression: 3] 2q)\(p — q — 1)! he 14 00 ag Piette ae ol Proof By employing Lagrangian inversion with two parameters, from (15) and (16) one 36 Shude Long and Junliang Cai may find that = (p,q) (1 + 8)P7*(1 = 26n) he(y, z) = = on ane eae (1+ 6)?-9-1(1 — 26n) = > ae ee e ag_90 t+ OFT =1,q-1) (1+ )P >t Ee = one On) anya >>) 3 ea mee i@ a\( —1)! (p—4 =1+ y? 24, Sha lq ie 2q)\(q — 1)! which is just the theorem. In what follows we present a corollary of Theorem 3.1. y? zt Corollary 3.1 The enumerating eee H = He(y) has the following explicit expression: (n — 2q—1)! n 0) =D ety Proof It follows immediately from (17) by putting x = y andn=p+4q. Now, let se cle) ~ 14800 By substituting (15) and (19) into Equ. (8), one may find that 1 = 1 — Sono ° 1+€0 (18) (19) (20) By (15), (19) and (20), we have the parametric expression of the function f = fe(x, y, z) as follows: _€(1 +8) oa OE Seep eg? 1 yz=n(l—On), f= | _ Sono” (1+€6)? According to (21), we have a 0 et ps : 7 1 —26n Gn) a > ~ (1+ €0)(1+0)(1 — 6)" 0 * — 1-0n (21) (22) 37 Enumeration of Rooted Nearly 2-Regualr Simple Planar Maps Theorem 3.2 The enumerating function f = fe(x,y,z) has the following explicit expression: (Ee) LJ p+q min{| 51,4} ap DS) eS p22 q=1 Il=3 k=max{1,[424-2]} —q—-1l+2k-1 a ot )atvret p—2q—1+3k Proof By using Lagrangian inversion with three variables, from (21) and (22) one may find that (2g—k—1)!k (q—k)!q! 1—-2k-1 fe(a, y, z | — 3k (23) Sa ‘(1 +0)P 4 (1 = 260) (x,y,z oe oF al yet a x4 l,p,q>0 (1 ae On)a+1 i = Se => alten aq) (1 + €0)' (1 + 0)? 41 (1 — 26m) alyP24 Gee +1 €20n (140)? l,p>0 q=0 (1 — 0n)2 1- oe min{ [$1], } =) P—-G4d iT Te (1+ 0)! 2k-1 a+ey d l,p>1 q=1 x (1 ene Se min{|$J,q} a4yy Dy l,p>1q=1 p= max({0, pits %¢,6,n) (1 — @n)at1 — 20n)a!yP 24 ( [—-2k-1 l— 3k 1} a _py (1+ 0)? 42k 1 (1 — 26 sé a. 14+2k,q ky ( +8) ( D ly? 24 min{|$],q} Cony [—2k-1 = 2 > a ( |— 3k ) l,p>1 q= Lp max{0,[ +=? ]} (0 on a = On)Ttt (p—q L+2k—1,q—k—1) (1 + 0)P- 94 2k1 Pate ae Gd —-op) vy’ z | p+q min{| 4] ah (Qqg—k—1)!k (l-2k-1 p22 q=1 I=3 p= max{1,[4 b2q— L42q—p yy x ee a a 1 aly? x4 J p+qa min{(4] ah (Q9g—k—1)!k (l—2k-1 p>2 q=1 1=3 p= max{1, pit2a=—p i P}} p—q—1l+2k—-1 alyP2!, p—2q—1+ 3k which is what we wanted. Finally, we give a corollary of Theorem 3.2. 38 Shude Long and Junliang Cai Corollary 3.2 The enumerating function g = ge(x,y) has the following explicit expression: n LB) min{|$1,q} 2qg—k- lk (l-2k-1 ge(t,y) =1+ > aera 13k ) n>3 l=3 q=1 k=max{1,[434-"}} nm—2q—l+2k—-1)\ , m 24 ( n—3q—-1+3k jay 28) Proof It follows soon from (23) by putting « = y andn=p+q. References 1 Junliang Cai and Yanpei Liu, A functional equation for enumerating simple Eulerian planar maps, J. Northern Jiaotong Univ., 21 (1997), 554-564. Yanpei Liu, A note on the number of loopless Eulerian planar maps, J. Math. Res. Expos., (12) 1992, 165-169. Yanpei Liu, Enumerating rooted simple planar maps, Acta. Math. Appl. Sinica (Eng. Ser.), 2 (1985), 101-111. 4] Yanpei Liu, The enumeration of simple planar maps, Utilitas Math., 34 (1988), 97-104. 5] Yanpei Liu, Enumerating rooted loopless planar maps, Acta Math. Appl. Sinica (Eng. Ser.), 2 (1985), 14-26. Yanpei Liu, On functional equations arising from map enumerations, Discrete Math., 123 (1993), 93-129. Yanpei Liu, An enumerating equation of simple planar maps with face partition (in Chi- nese), Acta Math. Appl. Sinica, 12(1989), 292-295. Yanpei Liu, On the enumeration of simple planar maps (in Chinese), Acta Math. Sinica, 31(1988), 279-282. Yanpei Liu, Functional equation for enumerating loopless Eulerian maps with vertex par- tition (in Chinese), Kerue Tongbao, 35(1990), 1041-1044. Yanpei Liu, On enumerating equation of planar maps (in Chinese), Adv. Math. (China), 18 (1989), 446-460. Yanpei Liu, Enumerative Theory of Maps, Kluwer, Boston, 1999. W.T.Tutte, A census of planar maps, Canad. J. Math., 15 (1963), 249-271. W.T.Tutte, A census of slicing, Canad. J. Math., 14 (1962), 705-722. Math. Combin. Book Ser. Vol.2(2012), 89-51 On Pathos Total Semitotal and Entire Total Block Graph of a Tree Muddebihal M. H. (Department of Mathematics, Gulbarga University, Gulbarga, India) Syed Babajan (Department of Mathematics, Ghousia College of Engineering, Ramanagaram, India) E-mail: babajan.ghousia@gmail.com Abstract: In this communication, the concept of pathos total semitotal and entire total block graph of a tree is introduced. Its study is concentrated only on trees. We present a characterization of graphs whose pathos total semitotal block graphs are planar, maxi- mal outerplanar, minimally nonouterplanar, nonminimally nonouterplanar, noneulerian and hamiltonian. Also, we present a characterization of those graphs whose pathos entire to- tal block graphs are planar, maximal outerplanr, minimally nonouterplanar, nonminimally nonouterplanar, noneulerian, hamiltonian and graphs with crossing number one. Key Words: Pathos, path number, Smarandachely block graph, semitotal block graph, total block graph, pathos total semitotal block graph, pathos entire total block graph, pathos length, pathos point, inner point number. AMS(2010): 05075 §1. Introduction The concept of pathos of a graph G was introduced by Harary [2], as a collection of minimum number of line disjoint open paths whose union is G. The path number of a graph G is the number of paths in a pathos. A new concept of a graph valued functions called the semitotal and total block graph of a graph was introduced by Kulli [5]. For a graph G(p,q) if B = U1, U2, U3,°°* ,Ur;7 > 2 is a block of G. Then we say that point u; and block B are incident with each other, as are ug and B and soon. If two distinct blocks B, and Bg are incident with a common cut point, then they are called adjacent blocks. The points and blocks of a graph are called its members. A Smarandachely block graph TY (G) for a subset V C V(G) is such a graph with vertices V U 6 in which two points are adjacent if and only if the corresponding members of G are adjacent in (V)q or incident in G’, where B is the set of blocks of G. The semitotal block graph of a graph G denoted T,(G) is defined as the graph whose point set is the union of set of points, set of blocks of G in which two points are adjacent if and only if 1Received March 26, 2012. Accepted June 12, 2012. 40 Muddebihal M. H. and Syed Babajan members of G are incident. The total block graph of a graph G denoted by Tg(G) is defined as the graph whose point set is the union of set of points, set of blocks of G in which two points are adjacent if and only if the corresponding members of G are adjacent or incident. Also, a new concept called pathos semitotal and total block graph of a tree has been introduced by Muddebihal [10]. The pathos semitotal graph of a tree T denoted by Pp, (T) is defined as the graph whose point set is the union of set of points, set of blocks and the set of path of pathos of T in which two points are adjacent if and only if the corresponding members of G' are incident and the lines lie on the corresponding path P; of pathos. The pathos total block graph of a tree T denoted by Pr,(T) is defined as the graph whose point set is the union of set of points, set of blocks and the set of path of pathos of T in which two points are adjacent if and only if the corresponding members of G are adjacent or incident and the lines lie on the corresponding path Pi of pathos. Stanton [11] and Harary [3] have calculated the path number for certain classes of graphs like trees and complete graphs. All undefined terminology will conform with that in Harary [1]. All graphs considered here are finite, undirected and without loops or multiple lines. The pathos total semitotal block graph of a tree T’ denoted by is defined as the graph whose point set is the union of set of points and set of blocks of T and the path of pathos of T in which two points are adjacent if and only if the corresponding members of T are incident and the lines lie on the corresponding path P; of pathos. The pathos entire total block graph of a tree denoted by is defined as the graph whose set of points is the union of set of points, set of blocks and the path of pathos of T in which two points are adjacent if and only if the corresponding members of T are adjacent or incident and the lines lie on the corresponding path P; of pathos. Since the system of pathos for T is not unique, the corresponding pathos total semitotal block graph and pathos entire total block graphs are also not unique. In Figure 1, a tree T and its semi total block graph T}(T) and their pathos total semitotal block graph are shown. In Figure 2, a tree T and its total block graph Tg(T) and their pathos entire total block graphs are shown. The line degree of a line wv in T, pathos length in T, pathos point in T’ was defined by Muddebihal [9]. If G is planar, the inner point number 7(G) of G is the minimum number of points not belonging to the boundary of the exterior region in any embedding of G in the plane. A graph G is said to be minimally nonouterplanar if i(G) = 1 as was given by Kulli [4]. We need the following results for our further results. Theorem A([10]) For any non-trivial tree T, the pathos semitotal block graph Pr,(T) of a tree T, whose points have degree d;, then the number of points are (2g +k+1) and the number of ie lines are (2 +2+ 5 D2 d? |, where k is the path number. i=1 Theorem B((10]) For any non-trivial tree whose points have degree d;, the number of points 12 and lines in total block graph Tp(T) of a tree T are (2g +1) and (2 + 5 S- a) ; i=l Theorem C((10]) For any non-trivial tree T, the pathos total block graph Pr,(T) of a tree On Pathos Total Semitotal and Entire Total Block Graph of a Tree 41 T, whose points have degree d;, then the number of points in Pr,(T) are (2g +k +1) and the P number of lines are (: +2+4+ S- a) , where k is the path number. i=1 T T,(T) Figure 1 Theorem D((7|) The total block graph Tp(G) of a graph G is planar if and only if G is outerplanar and every cut point of G lies on at most three blocks. Theorem E((6]) The total block graph Tp(G) of a connected graph G is minimally nonouter- 42 Muddebihal M. H. and Syed Babajan planar if and only if, (1) G “éf a cycle, or (2) G is a path of length n > 2, together with a point which is adjacent to any two adjacent points of P. Theorem F'((8]) The total block graph Tp(G) of a graph G has crossing number one if and only af, (1) G ts outerplanar and every cut point in G lies on at most 4 blocks and G has a unique cut point which lies on 4 blocks, or (2)G is minimally nonouterplanar, every cut point of G lies on at most 3 blocks and exactly one block of G is theta-minimally nonouterplanar. Corollary A((1]) Every non-trivial tree T contains at least two end points. §2. Pathos Total Semitotal Block Graph of a Tree We start with a few preliminary results. Remark 2.1 The number of blocks in pathos total semitotal block graph P;,,(T') of a tree T is equal to the number of pathos in T. Remark 2.2 If the pathos length of the path P; of pathos in T is n, then the degree of the corresponding pathos point in P.(T) is 2n + 1. In the following theorem we obtain the number of points and lines in pathos total semitotal block graph P;,)(T) of a tree T. Theorem 2.1 For any non-trivial tree T, the pathos total semi total block graph Prs»(T) of a tree T, whose points have degree d;, then the number of points in Pisy(T) are (2g+k+1) and the number of lines are i=l doy os (% +24 3 y a) where k is the path number. Proof By Theorem A, the number of points in Pr,(T) are (2g+k+ 1), and by definition of Ps)(Z'), the number of points in P;s,(T') are (24g+k +1), where k is the path number. Also 12 by Theorem A, the number of lines in Pr,(T) are | 2q¢+2+ 5 S- d? |). The number of lines i=1 in P:s)(T') is equal to the sum of lines in Py,(T) and the number of lines which lie on the lines (or blocks) of pathos, which are equal to q, since the number of lines are equal to the number of blocks in a tree T. Thus the number of lines in P;,,(T') is equal to ee os ee q+ Qq+2+5 > di )] =r 24 $50" i=1 i=l On Pathos Total Semitotal and Entire Total Block Graph of a Tree 43 §3. Planar Pathos Total Semitotal Block Graphs A criterion for pathos total semitotal block graph P;,,(T) of a tree T to be planar is presented in our next theorem. Theorem 3.1 For any non-trivial tree T, the pathos total semi total block graph Prs»(T) of a tree T is planar. Proof Let T be a non-trivial tree, then in T,(T') each block is a triangle. We have the following cases. Case 1 Suppose G is a path, G = Pn : u,U2,U3,°+:,Un,n > 1. Further, V[T%(T)] = {ui, U2, U3,°°* ; Un, b1, b2, bs,-+- ,bn—i}, where 61, be, b3,--+ ,bn—1 are the corresponding block points. In pathos total semi total block graph P;..(T') of a tree T, the pathos point w is adjacent to, {u1, U2, U3,---,Un, b1, ba, bs,--- ,bn-1}. For the pathos total semitotal block graph Pisp(L) of a tree T, {urd uow, ugbousw, ugb3uaw, +++ ,Un—1bn-1Unw} © V[Piso(L)], in which each set {Un—1bn—-1Unw} forms an induced subgraph as Ky. Hence one can easily verify that each induced subgraphs of corresponding set {u,—1b,-1Unw} is planar. Hence P;,»(T) is planar. Case 2 Suppose G is not a path. Then V [T7,(G)] = {u1, ue, us,--+ , Un, b1, be, b3,-++ , bn—1} and W1, W2, W3,°** , Wr be the pathos points.Since uy,—1Un is a line and Up—1Un = bn—1 € V [T)(G)]. Then in P:.)(G) the set {tn—1, bn—1, Un, w} Vn > 1, forms Ky as an induced subgraphs. Hence Pis5p(T) is planar. The next theorem gives a minimally nonouterplanar P,,.(T). Theorem 3.2 For any non-trivial tree T, the pathos total semitotal block graph Pis)(T) of a tree T is minimally nonouterplanar if and only if T = Ko. Proof Suppose T = K3, and P;.,(T') is minimally nonouterplanar, then T,(T) = K4 and one can easily verify that ¢(P:..(T')) > 1, a contradiction. Suppose T #4 K2. Now assume T = Ky 2 and Pys)(T) is minimally nonouterplanar. Then T)(T) = kz -k3. Since Ky,2 has exactly one pathos and let v be a pathos point which is adjacent to all the points of ks - ks in Pisy(T'). Then one can easily see that, i (Pis)(T)) > 1a contradiction. For converse, suppose T = Ko, then T,(T) = K3 and Prsy(T) = K4. Hence Pis4(T) is minimally nonouterplanar. From Theorem 3.2, we developed the inner point number of a tree as shown in the following corollary. Corollary 3.1 For any non-trivial tree T with q lines, i (Pisp(T)) = ¢. Proof The result is obvious for a tree with g = 1 and 2. Next we show that the result is true for g > 3. Now we consider the following cases. Case 1 Suppose T is a path, P : vy, v2,...,Un such that vyvo = e1,v2V3 = €2°++ 4 Un—1Un = 44 Muddebihal M. H. and Syed Babajan €n—1 be the lines of P. Since each e;,1 < i < n—1, be a block of P, then in T,(P), each e; is a point such that V[T,(P)] = V(P)U E(P). In T,(P) each v1 e1v2, v3e203,° ++ , Un—1€n—1Un forms a block in which each block is k3. Since each line is a block in P,then the number of ks’s in Tb(P) is equal to the numbers of lines in P. In P;s4(P), it has exactly one pathos. Then V[Piso(P)] = V[Tp(P)] U {P} and P together with each block of T,(P) forms a block as Ps)(P). Now the points p, v1, €1, v2 forms k4 as a subgraph of a block P;,,(P). Similarly each {v2, €2, v3, D}, {v3, e3, V4, P},+++ » {Un—1, Cn—1, Un, p} forms ka as a subgraph of a block P,.,(P). One can easily find that each point e;,1 <7 < n-—1 lie in the interior region of ky, which implies that 7 (Prso(P)) = q. Case 2 Suppose T is not a path, then T' has at least one point of degree greater than two. Now assume T' has exactly one point v, degu > 3. Then T = K1,». If Prs)(T) has inner point number two which is equal to n = q. Similarly if n is odd then P,,,(T’) has n— 1 blocks with inner point number two and exactly one block which is isomorphic to k4. Hence ¢ [Pis5 (1.n)] = q. Further this argument can be extended to a tree with at least two or more points of degree greater two. In each case we have ¢[Pisp (T')] = ¢. In the next theorem, we characterize the noneulerian P;.4(T). Theorem 3.3 For any non-trivial tree T, the pathos total semitotal block graph Pis,(T) of a tree T is noneulerian. Proof We have the following cases. Case 1 Suppose A(T) < 2 and if p = 2 points, then P;,,(1) = K4, which is noneulerian. If T is a path with p > 2 points. Then in T,(T)) each block is a triangle such that they are in sequence with the vertices of Tb(T) as {v1, b1, v2, v1} an induced subgraph as a triangle in T;,(T). Further {v2, be, v3, va}, {u3, 03, U4, U3},°°+ 5 {Un—1,; bn, Un, Un—1}, in which each set form a triangle as an induced subgraph of T,(T'). Clearly one can easily verify that T,(T) is eulerian. Now this path has exactly one pathos point say k,, which is adjacent to v1, v2, U3,°++ , Un, 61, b2, b3,°°° ,bn—1 in Pis5)(L) in which all the points v1, v2, v3, ..., Un, b1, be, b3,-++ ,bn—-1 © Pisy(L) are of odd degree. Hence P;,4(T’) is noneulerian. Case 2 Suppose A(T) > 3. Assume T has a unique point of degree > 3 and also assume that T = Ky. Then in 7;(T) each block is a triangle, such that there are n number of blocks which are K3 with a common cut point k. Since the degree of a vertex k = 2n. One can easily verify that Tp (41,3) is eulerian. To form P;.,(T'), T = Kin, the points of degree 2 and the point k are joined by the corresponding pathos point which gives points of odd degree in P;,,(T'). Hence P,s)(T) is noneulerian. In the next theorem we characterize the hamiltonian P;.,(T). Theorem 3.4 For any non-trivial tree T, the pathos semitotal block graph Prs)(T) of a tree T is hamiltonian if and only if T is a path. Proof For the necessity, suppose T is a path and has exactly one path of pathos. Let V [Tp (T)] => {ui, U2,U3,°°° ,Un}{bi, ba, bs, sneer sOn-1}, where bi, ba, bs, seey bn—1 are On Pathos Total Semitotal and Entire Total Block Graph of a Tree 45 block points of T. Since each block is a triangle and each block consists of points as By = {ui, bi, u2}, Bo = {ua, be, us},---,Bm = {Um,bm,Um+i}. In Prsy(T) the pathos point w is adjacent to {w1, U2, Us,°** » Un, 01, be, b3,-++ ,bn—1}. Hence V [Pish(T)] = {ur, ue, g,+++ , Un} U {b1, bo, bs, +++ ,bn—1} Uw form a spanning cycle as w, w1, b1, U2, b2, W2,°+* ,Un—1, 0n—1, Un, w of Pis)(L). Clearly P:;)(T) is hamiltonian. Thus the necessity is proved. For the sufficiency, suppose P;.,(T') is hamiltonian. Now we consider the following cases. Case 1 Assume T is a path. Then T has at least one point with degu > 3, V v € V(T), suppose TJ’ has exactly one point u such that degree u > 2 and assume G = T' = Ky,,,. Now we consider the following subcases of case 1. Subcase 1.1 For Ki,,,n > 2 and if n is even, then in T}(T) each block is kg. The number of path of pathos are = Since n is even we get ie blocks in P;s5(T'), each block contains two times of (A4) with some edges common. Since Pisy(T) has a cut points, one can easily verify that there does not exist any hamiltonian cycle, a contradiction. 1 Subcase 1.2 For Ky,,n > 2 and n is odd, then the number of path of pathos are “ , We asaploees ahh = is nonline disjoint subgraph of P;,,(T) and remaining blocks is (K4). Since P,,,(T') contain since n is odd we get blocks contains two times of (f(4) which a cut point, clearly P;,,(T) does not contain a hamiltonian cycle, a contradiction. Hence the sufficient condition. §4. Pathos Entire Total Block Graph of a Tree A tree T, its total block graph Tg (T), and their pathos entire total block graphs P.1)(T) are shown in Figure 2. We start with a few preliminary results. Remark 4.1 If the pathos length of path P; of pathos in T is n, then the degree of the corresponding pathos point in P.(T) is 2n + 1. Remark 4.2 For any nontrivial tree T’, the pathos entire total block graph P.1»(T) is a block. Theorem 4.1 For any non-trivial tree T, the pathos total block graph P.1»(T) of a tree T, whose points have degree d;, then the number of points in Pe(T) are (2g+k+1) and the number of P lines are (2 +2+4+ SB a) , where k is the path number. i=1 Proof By Theorem C, the number of points in Pr,(T)are (2g+k+ 1), by definition of Pe»(T), the number of points in P.4,(T) are (2g+k-+1), where k is the path number in T. ie Also by Theorem B, the number of lines in Tg(T) are | 2g + 5 S- ie) . By Theorem C, The i=1 Pp number of lines in Pr,(T) are | q+2+ d? |. By definition of pathos entire total block 'B i Ma i=l graph P.1,(T) of a tree equal to the sum of lines in Pp, (T) and the number of lines which lie on block points b; of Tg(T) from the pathos points P;, which are equal to g. Thus the number 46 Muddebihal M. H. and Syed Babajan Pp P of lines in Po (T) = (s+24 4] = (+24 oa), i=1 i=l T,(T) Figure 2 §5. Planar Pathos Entire Total Block Graphs A criterion for pathos entire total block graph to be planar is presented in our next theorem. Theorem 5.1 For any non-trivial tree T, the pathos entire total block graph P.1»(T) of a tree T is planar if and only if A(T) < 3. Proof Suppose P.(T) is planar. Then by Theorem D, each cut point of T lie on at most 3 blocks. Since each block is a line in a tree, now we can consider the degree of cut points instead of number of blocks incident with the cut points. Now suppose if A(T) < 3, then by Theorem D, Tg(T) is planar. Let {b1, 62, b3,--- ,bp-1} be the blocks of T with p points such that b) = e1, bz = €2,--- ,bp_1 = €p_1 and P; be the number of pathos of T. Now V [Pew(T)| = V(G) VU b1, ba, 63,-++ ,bp-1 U{P;}. By Theorem D, and by the definition of pathos, On Pathos Total Semitotal and Entire Total Block Graph of a Tree 47 the embedding of P.:»(T) in any plane gives a planar P.1,(T). Conversely, Suppose A(T) > 4 and assume that P.4,(T) is planar. Then there exists at least one point of degree 4, assume that there exists a vertex v such that degu = 4. Then in Tp(T), this point together with the block points form ks as an induced subgraph. Further the corresponding pathos point which is adjacent to the V(T’) in Tg(T) which gives P.i»(T) in which again ks as an induced subgraph, a contradiction to the planarity of Pe1)(T). This completes the proof. The following theorem results the minimally nonouterplanar P.+5(T). Theorem 5.2 For any non-trivial tree T, the pathos entire total block graph P.1»(T) of a tree T is minimally nonouterplanar if and only if T = ka. Proof Suppose T = kg and Per»(T) is minimally nonouterplanar. Then Tp(T) = kg and one can easily verify that, i(P.1(T')) > 1, a contradiction. Further we assume that T = Ki,2 and P.4)(T) is minimally outerplanar, then Tg(T) is W, — x, where x is outer line of W,. Since F1,2 has exactly one pathos, this point together with W, — x gives W,+1. Also in P.1,(T’) and by definition of P.i»(T) there are two more lines joining the pathos points there by giving W,+3. Clearly, P.1,(T) is nonminimally nonouterplanar, a contradiction. For the converse, if T = ko, T p(T) = kg and P.4(T) = K4 which is a minimally nonouter- planar. This completes the proof of the theorem. Now we have a pathos entire total block graph of a path p > 2 point as shown in the below remark. Remark 5.1 For any non-trivial path with p > 2 points, i[P.1,(T)| = 2p—3. The next theorem gives a nonminimally nonouterplanar P.1(T). Theorem 5.3 For any non-trivial tree T, the pathos entire total block graph P.1»(T) of a tree T is nonminimally nonouterplanar except for T = ka. Proof Assume T is not a path. We consider the following cases. Case 1 Suppose T is a tree with A(T) > 3. Then there exists at least one point of degree at least 3. Assume v be a point of degree 3. Clearly, T = K,,3. Then by the Theorem F, i(Tp(T)] > 1. Since Tp(T) is a subgraph of P.1»(T'). Clearly, i (Peiw(T)) > 2. Hence Pery(T) is nonminimally nonouterplanar. Case 2 Suppose T is a path with p points and for p > 2 points. Then by Remark 5.1, t[Pew(T)| > 1. Hence Peyy(T) is nonminimally nonouterplanar. In the following theorem we characterize the noneulerian P.4y(T). Theorem 5.4 For any non-trivial tree T, the pathos entire total block graph P.1»(T) of a tree T is noneulerian. Proof We consider the following cases. 48 Muddebihal M. H. and Syed Babajan Case 1 Suppose T' is a path P, with n points. Now for n = 2 and 8 points as follows. For p = 2 points, then P.4,() = K4, which is noneulerian. For p = 3 points, then P.,(T) is a wheel Wg together with two lines joining the non adjacent points in which one point is common for these two lines as shown in the Figure 3, which is noneulerian. b, b, eo P, L b, b, P, P.(T) Figure 3 For p > 4 points, we have a path P : v1, v2, v3,...,Up. Now in path each line is a block. Then viva = e1 = bi, v2vg = €2 = be,...,Up-1Up = Ep-1 = bp-1, V €p-1 € E(G), and V bp-1 € V[Tp(P)]. In Tp (P), the degree of each point is even except b; and 6,1. Since the path P has exactly one pathos which is a point of P..»(P) and is adjacent to the points V1, V2, U3,---,Up, Of Tg (P) which are of even degree, gives as an odd degree points in Pery(P) including odd degree points b; and bp)_1. Clearly P.i,(P) is noneulerian. Case 2 Suppose T is not a path. We consider the following subcases. Subcase 2.1 Assume T has a unique point degree > 3 and T = Ky, with n is odd. Then in Tp (T) each block is a triangle such that there are n number of triangles with a common cut points k which has a degree 2n. Since the degree of each point in Tg (K1,,) is odd other than the cut point k which are of degrees either 2 or n +1. Then Pe4,(T) eulerian. To form P.1(T) where T = Kj, the points of degree 2 and 4 the point k are joined by the corresponding pathos point which gives (2n + 2) points of odd degree in Perp(Kin). Pet(T) has n points of odd degree. Hence P.1,(T) noneulerian. Assume that T = Ky,n, where n is even, Then in Tg (T) each block is a triangle, which are 2n in number with a common cut point k. Since the degree of each point other than k is either 2 or (n +1) and the degree of the point & is 2n. One can easily verify that Tp (Kin) is noneulerian. To form Per»(T) where T = Ky,n, the points of degree 2 and 5 the point & are joined by the corresponding pathos point which gives (n + 2) points of odd degree in Peyy(T). On Pathos Total Semitotal and Entire Total Block Graph of a Tree 49 Hence P.+(T) noneulerian. Subcase 2.2 Assume T has at east two points of degree > 3. Then V[Tp(T)] = V(G) U bi, b2, b3,...,bp, V bp € E(G). In Tg (T), each endpoint has degree 2 and these points are adjacent to the corresponding pathos points in P.4,(T) gives degree 3, From case 1, Tree T has at least 4 points and by Corollary [A], Pei(T) has at least two points of degree 3. Hence P.1»(T) is noneulerian. In the next theorem we characterize the hamiltonian P.4,(T). Theorem 5.5 For any non-trivial tree T, the pathos entire total block graph P.1»(T) of a tree T is hamiltonian. Proof we consider the following cases. Case 1 Suppose T is a path with {uj, ue, us,...,Un} € V(T) and 64, be, b3, ...,bm be the number of blocks of T such that m = n—1. Then it has exactly one path of pathos. Now point set of Tp (T), V [Ta(T)] = {ur,u2,--: , un} U {b1, bo,...,bm}. Since given graph is a path then in Tg (T), 61 = e1,b2 = €2,...,bm = Cm, such that by, be, bs,...,0m C V [Te (T)]. Then by the definition of total block graph, {u1, u2,...,Um—1, Um} U {b1, b2,.--,bm—1, bm} U {byu1, bou2,...-bmUn—1, mtn} form line set of Tg (T) (see Figure 4). b, b, b, b,, P e e ° SSeS eee) u, Uy Us Uy Uy U,, b, b, b, b,, na: “~~ N\LZN LS U, Uy Us Uy Uy U, Figure 4 Now this path has exactly one pathos say w. In forming pathos entire total block graph of a path, the pathos w becomes a point, then V [P.15(T)] = {u1, ua,-++ ,Un}U{b1, ba,...,bm}U{w} and w is adjacent to all the points {u1,u2,--- , U,} shown in the Figure 5. In P.»(T), the hamiltonian cycle w, ur, 01, Ua, be, U2, Us, b3,°++ ,Un—1, 0m; Un, w exist. Clearly the pathos entire total block graph of a path is hamiltonian graph. Case 2 Suppose T is not a path. Then T has at least one point with degree at least 3. Assume that T has exactly one point u such that degree > 2. Now we consider the following subcases of Case 2. Subcase 2.1 Assume T = Kj,,,n > 2 and is odd. Then the number of paths of pathos are 1 is Let V [Tp(T)] = {ui, u2,..-,Un, 01, b2,-.., bmi}. By the definition of pathos total block graph. By the definition P.w(T) V [Pew(T)] = {u1, ua,.--, Un, b1, b2,.-.,bn—-1} U {pi, pa, - ;Pn+1/2}- Then there exists a cycle containing the points of By the definition of P.4,(T) as 50 Muddebihal M. H. and Syed Babajan pi, U1, b1, be, U3, pa, U2, b3, U4,+*+ , pi and is a hamiltonian cycle. Hence P.:»(T) is a hamiltonian. Figure 5 Subcase 2.2 Assume T = Kj,,,n > 2 and is even. Then the number of path of pathos are - then V [Tp(T)] = {u1, u2,-++ Un, b1, b2,-++ ,bn-1}. By the definition of pathos en- tire total block graph P..(T) of a tree T. V [Pes(T)] = {ur, ua,--+ , Un, b1, b2,--+ ,bn—1} U {P1;P2;*** ;Pn/2}. Then there exist a cycle containing the points of Pero(T) as pi, us, b1, be, U3, P2, U4, bs, b4,--+ , pi and is a hamiltonian cycle. Hence P.4,(T) is a hamiltonian. Suppose 7’ is neither a path nor a star, then T contains at least two points of degree > 2. Let u1,uU2,uUg,°---,Un be the points of degree > 2 and vj, v2,v3,:::,Um be the end points of T. Since end block is a line in T, and denoted as bj, b2,--- ,b, ,then V [Tp(T)] = {ui, U2,---,Un} U {01, v2,...Um} U {b1, be,..., be },and tree T has p; pathos points, i > 1 and each pathos point is adjacent to the point of T where the corresponding pathos lie on the points of T. Let {pi,pe2,--- ,pi}be the pathos points of T. Then there exists a cycle C containing all the points of P.w(T), as pr, v1, b1, v2, po, U1, b3, U2, D3, U3, b4,; Um—1; On—1, bn, Um;---, Pi- Hence P.1»(T) is a hamiltonian cycle. Clearly, P..»(T') is a hamiltonian graph. In the next theorem we characterize P.4,(T) in terms of crossing number one. Theorem 5.6 For any non-trivial tree T, the pathos entire total block graph P.1»(T) of a tree T has crossing number one if and only if A(T) < 4, and there exist a unique point in T of degree 4. Proof Suppose P.1,(T') has crossing number one. Then it is nonplanar. Then by Theorem 5.1, we have A(T’) < 4. We now consider the following cases. Case 1 Assume A(T’) = 5. Then by Theorem [F], Tg (T) is nonplanar with crossing number more than one. Since Tg (T) is a subgraph of Pr, (T). Clearly cr (Pr, (T)) > 1, a contradiction. Case 2 Assume A(T) = 4. Suppose T has two points of degree 4. Then by Theorem F, Tz (T) has crossing number at least two. But Tg (T’) is a subgraph of P.»(T). Hence cr (Pew(T)) > 1, On Pathos Total Semitotal and Entire Total Block Graph of a Tree 51 a contradiction. Conversely, suppose T’ satisfies the given condition and assume T' has a unique point v of degree 4. The lines which are blocks in T such that they are the points in Tg (T). In Tg (T), these block points and a point v together forms an induced subgraph as ks. In forming P..»(T), the pathos points are adjacent to at least two points of this induced subgraph. Hence in all these cases the cr (P.1,(T')) = 1. This completes the proof. References [1] Harary F., Graph Theory, Addison-Wesley Reading. Mass. (1969). [2] Harary F., Covering and packing in graphs-I, Annals of Newyork Academy of Sciences, [3 (1970), pp 198-205. Harary F. and Schweak A.J., Evolution of the path number of graph, Covering and pack- ing in graph-II, Graph Theory and Computing, Ed.R.C. Read, Academic Press, Newyork (1972), pp 39-45. Kulli V.R., On minimally nonouterplanar graphs, Proceeding of the Indian National Science Academy, Vol-41, Part-A, No.3, (1975), pp 275-280. Kulli V.R., The semi total-block graph and Total-block graph of a graph, Indian J. Pure €& App. Maths, Vol-7, No.6, (1976), pp 625-630. Kulli V.R. and Patil H.P., Minimally nonouterplanar graphs and some graph valued func- tions, Journal of Karnataka University Science, (1976), pp 123-129. Kulli V.R. and Akka D.R., Transversability and planarity of total block graphs, Jour. Math. Phy. Sci., Vol-11, (1977), pp 365-375. Kulli V.Rand Muddebihal M.H., Total block graphs and semitotal block graphs with cross- ing numbers, Far East J. Appl. Math., 4(1), (2000), pp 99-106. Muddebihal M.H., Gudagudi B.R. and Chandrasekhar R., On pathos line graph of a tree, National Academy Science Letters, Vol-24, No.5 to 12, (2001), pp 116-123. Muddebihal M.H. and Syed Babajan, On pathos semitotal and total block graph of a tree (to appear). Stanton R.G., Cowan D.D. and James L.O., Some results on the path numbers, Preoceed- ings of Lousiana Conference on Combinatorics, Graph Theory and Computing, (1970), pp 112-135. Math.Combin. Book Ser. Vol.2(2012), 52-58 On Folding of Groups Mohamed Esmail Basher (Department of Mathematics, Faculty of Science in Suez, Suez Canal University, Egypt) E-mail: m_e_-basher@yahoo.com Abstract: The aim of our study is to give a definition of the folding of groups and study the folding of some types of groups such as cyclic groups and dihedral groups, also we discussed the folding of direct product of groups. Finally the folding of semigroups are investigated. Key Words: Folding, multi-semigroup, multi-group, group, commutative semigroup. AMS(2010): 54005, 20K01 §1. Introduction In the last two decades there has been tremendous progress in the theory of folding. The notion of isometric folding is introduced by 5S. A. Robertson who studied the stratification determined by the folds or singularities [10]. The conditional foldings of manifolds are defined by M. El- Ghoul in [8]. Some applications on the folding of a manifold into it self was introduced by P. Di. Francesco in [9]. Also a folding in the algebras branch introduced by M.El-Ghoul in [7]. Then the theory of isometric foldings has been pushed and also different types of foldings are introduced by E. El-Kholy and others [1,2,5,6]. Definition 1.1([4]) A non empty set G on which is defined s > 1 associative binary operations *x is called a multi-semigroup, if for all a,b € G,ax*b © G, particularly, if s = 1, such a multi-semigroup is called a semigroup. Example 1 Z, = {0,1,2,...,p — 1} is a semigroup under multiplication p, p € Zt. Definition 1.2([4]) A subset H is a subsemigroup of G if H is closed under the operation of G ; that it ifaxbe H for alla,be H. Definition 1.3 A multi-group (G,O) is a non empty set G together with a binary operation set O onG such that for * € O, the following conditions hold: (1) Va,be G thenaxbeEG. (2) There exists an element e € G such thataxe =exa=a, for allaeG. 1 1 (3) Fora €G there is an element a~! in G such thataxa~'=a7!*xa=e. Particularly, if |O| = 1, such a multi-group is called a group and denoted by G lReceived January 17, 2012. Accepted June 14, 2012. 2Current Address: Qassim University,College of Science, Dept. of Math., P. O. Box 6644, Buriedah 51452, Al Qassim, Saudi Arabia On Folding of Groups 53 A group G is called Abelian if ax b = bx a for all a,b € G. The order of a group is its cardinality, i.e., the number of its elements. We denote the order of a group G by |G}. Definition 1.4([3]) The group G is called a cyclic group of order n, if there exists an element g € G, such that G = (g) = {g |g” = 1}. In this case g is called a generator of G. Definition 1.5([3]) The dihedral group Don, of order 2n, is defined in the following equivalent ways: Don = {a,b ae oP S41 ab = a}. Definition 1.6([3]) A subset H is a subgroup of G ,H < G, if H is closed under the operation of G ; that it ifa*xbe A for alla,be H. Definition 1.7([3]) The trivial subgroup of any group is the subgroup {e}, consisting of just the identity element. Definition 1.8([3]) A subgroup of a group G that does not include the entire group itself is known as a proper subgroup, denoted by H <G. Theorem 1.1 Every subgroup of a cyclic group is cyclic. §2. Group Folding In this section we give the notions of group folding and discuss the folding of some kinds of groups Definition 2.1 Let G,,Gz are two groups, The group folding g.f of Gi into G2 is the map f : (Gi, *) (Go, 0) st. VaeG, fla)=b, fla )=o! where b € Gg and f(G1) is subgroup of Go. Definition 2.2 The set of singularities \> f is the set of elements a; € G such that f(a;) = a;. Definition 2.3 A group folding is called good group folding g.g.f if H is non trivial subgroup of G. Proposition 1.1 The limit of group folding of any group is {e}. Proof Let G be any group and since any group has two trivial subgroups (G, *), ({e} , *) ,where ({e},*) is the minimum subgroup of G. Then if we define a sequence of group folding fi: (G,*) — (G,*), such that f(a;) =); , f(a;+) =b;*. We found that lim f;(G) = {e}. Theorem 2.1 Let G be cyclic group G = {g:g? =1}, where p is prime. Then there is no g-g-.f map can be defined on G. Proof Given G is cyclic group of order p, since p is prime. Then there is no proper subgroup can be found in G. Hence we can not able to defined any g.g.f map on group G. 54 Mohamed Esmail Basher 1 {XQ Theorem 2.2 Let G be cyclic group G = {g: g? =1}, where p = qf" q5?---¢0" and q, G2,°+* 4 dn are distinct prime number. Then the limit of g.g.f map of G is the proper subgroup , H = Pp {9% : (g#)* =1}. a1 {a2 Proof Since G be cyclic group and |G] = p = q["4q5?--- 2", %,425°°*5%n are distinct prime number. Then there exist proper subgroup H < G such that |H| = k. So we can defined g.g.f map fi :G— G, such that f\(G) = H, after this there exist two cases. Case 1. _ If k is prime number then we can not found a proper subgroup of H and so lim fi(G@) =H and k= ql Si<n, f(G) =H = {o% : (gh) * = 1h. Case 2. Ifk is not prime. Then k = qj'"q5? ---q%",m <n, hence there exist a proper subgroup Aer, A| =k. Thus we can defined a g.g.f map f2 : H —> H, such that fo(H) = H. Again if |#| is prime. Then lim f;(G) =H, k=q@,1<i<m, fi(G) =H = {9% : (g%)% = i} or, if |H | =k is not Srinie. Then we can repeat the Case II again. Finally the limit of g.g.f of G is the subgroup H = {9% : (g%)% = 1} : Corollary 2.1 If p= q®% then the limit of g.g.f map of G is a subgroup H = {9° : (9%) = i} : Example 2 Let G = {g:9'!2 =1} bea cyclic group, |G| = 12 = 27.3, so we can defined a g.g.f map as the following f; : G — G, such that: fi) =1, Ag’) = 97, Ag’) = 9”, filo) =94, A) =95, Ag®) =9", fig’) =97, fg’) =o", fig?) =9*, Ao?) =9°, filg) = 9°, Ai(g’’) = 9° and f1(G) = H = {1,9?,9*, 9°, 98, g'°}, since the order of H is not prime then we can defined ag.g.f map as the following f2 : H —> H such that fo(1) = 1, fa(g°) =1 and fo(H) = H = {1,9*, 98} is proper subgroup of H. Since the order of H is prime, then lim fi(G) = H = {9'\(9*)? = 1}. Proposition 2.2 For any dihedral group Do, = {a,b |a = b" = 1, bab= 1}, we can defined a g-g-.f map. a1 {a2 Theorem 2.3 Let Do, be a dihedral group ,where n = qy* qo? ++: ae™ and q1,92,°°* Um are distinct prime number. Then the limit of g.g.f of Dan is one of these H; = {1, ab'} = 1 Dect Sade Hi {vs (bu) = bi ae year On Folding of Groups 55 Proof Let Don = {a,b |a = b” = 1, bab = 1} be a dihedral group of order 2n. Then the group D2, has n proper subgroups H; = {1,ab'} ,i = 1,2, ...,n of order 2, and so we can defined ag.g.f map f : Den, —> Don, such that f(Don) = Hi. Also since the dihedral group has proper cyclic subgroup H = {b|b” = 1}. Then we can defined a g.g.f map f : Don —> Don, such that f(De,) = {b|b” = 1}, and since the subgroup H = {b|b” = 1} be cyclic group of order n = gf g5?--- qo. Thus by applying the Theorem 2.2 we get that lim fi(Dan) =H = {ov (bv) a = i} i= 1,2,-++,m. §3. Group Folding of the Direct Product of Groups In this section we discuss the group folding of direct product of groups. Let fi : (Gi,*) > (Gi,*), fo : (Go,*) (Go, *) are two g.g.f maps. Then we define the direct product of the fi, fa as the following: fi x fo : (Gy x G2,*) =? (Gi x G2,*) Va €Gi,b€ Go, (fi x fe) (a,) = (fila), fo(0)) Theorem 3.1 The direct product of two g.g.f maps are g.g.f map also, but the converse is not always true. Proof Since fy : (Gi,*) — (Gi,*), fo : (Go,*) — (Go,*) are two g.g.f maps then there exist two proper subgroups Hy, H2 of G1, G2 respectively, such that f1(Gi) = M1, fo(G2) = Ho. As the direct product of the any two proper subgroups are proper subgroups. Hence Hy; x Hp < G x Gp. Now we will proof that the map, f* = fi x fo: Gi x Gz — G, Xx G2 is g.g.f map. Let a,b € G1, G2 and c,d € Hj, H2 respectively, then we have: f* (a,b) = (fi x fa) (a,b) = (fila), f(b) = (Gd) € A x Aa, ft fabs") = (fi x fa) (a 1b ig) = (c ad *) E Ay x Ao. Hence the map f* is g.g.f map. To proof the converse is not true, let G; = {91 o2= 1} G2 = {go | B= 1} be cyclic groups and since the order of them are prime then from Theorem 2.1, we can not able to define a g.g.f map of them. But the direct product of G1, G2 is Gi x G2 = {(1, Ly, (1, 92), (1, 93), (1, 1), (91,92), (91,95) ¢ and since the |(G; x G2)| = 6. Hence G; x Gz has a proper subgroup H = {(1,1), (1, g2)} and so we can define the g.g.f map f* : G1 x Gz — G, x G2 as the following: FG, 1) = G, 1), f*(1, g2) = 1,92), F*(g1,1) = (1, 1), f*(g1, 92) = (1, 92), f*(91,.92) = (A, 92): This completes the proof. 56 Mohamed Esmail Basher §4. Folding of Semigroups In this section we will be discuss the folding of semigroups into itself. Let (G,*«) be a commu- tative semigroup with identity 1, i.e. a monoid. Definition 4.1([4]) A nonzero element a of a semigroup G is a left zero divisor if there exists a nonzero b such thata*xb=0. Similarly, a is a right zero divisor if there exists a nonzero element cE G such that cx a= 0. The element a is said to be a zero divisor if it is both a left and right zero divisor. we will denote to the set of all zero-divisors by Z(G), and the set of all elements which have the inverse by I(G). Definition 4.2 The zero divisor folding of the semigroup G, z.d.f, is the map fz : (G,*) > (G,x*), st. 0 ifx=0 filz)=4 a ifxexa=0, 27,a40 x ifxxa#0, r1,a40 where a € Z(G). Note that f,(G) may be semigroup or not. We will investigate the zero divisor folding for Zy semigroups. Definition 4.3 Let Zp be a semigroup under multiplication modulo p. The z.d.f map of Zp is the map fz, : (G,.) > (G,.), st. 0 ifx=0 © ifxq #0, xq #0 where q € Z(Zy ), is the greatest divisor of p. Proposition 4.1 If the order of Z, is prime. Then f,(G) =G, i.e. fz is identity map. Proof Since the order of semigroup Z, is prime. Then the semigroup Z, has not got any zero divisor, and so the z.d.f which can defined on Z, the identity map f,(x) = 2, for all LE Zp. Theorem 4.1 Let Zpbe semigroup of order p, then z.d.f of Zp into itself is a subsemigroup under multiplication modulo p. Has one zero divisor if 4| p or has not any zero divisor if 4{ p. Proof Let Z, be semigroup under multiplication modulo p. Then Z, consists of two subsets Z(Zp), I(Zp). On Folding of Groups 57 Case 1. If p is even, then the z.d.f map defined as follows: 0 ifs =0; f.(~)=4 § ifee Z(Z,), xis even; ifx€ Z(Z,), xis odd or z € I(Zy), 8 where 5 is the greatest divisor of p. Hence f,(Zp) = I(Zp)U4 0, S U1,°°° stn}, where £1,°°- ,2n are odd zero divisors and x;.2; # 0, for all i,j = 1,---,n. Notice that f,(Z, ) = I(Zp) U {0, = X1,°*+,4np is subsemigroup under multiplication modulo p. If 4 | p then a5 = 0, hence the subsemigroup f,(Z,) has one zero divisor = Otherwise, if 4 { p then a . #0. And so H =I1(Z,)U {0, a1, vee ,an is subsemigroup under multiplication modulo p without any zero divisor. Case 2. If p is odd, then the z.d.f map defined as follows: 0 ifx=0; f(t)=4 q¢ ifee Z(Z,), zis odd; x ifxé Z(Zp), x is even or x € I(Z,), where q is the greatest divisor of p. Hence f,(Z,) = I(Zp)U{0,q,21,--: ,2n}, where x1,--- ,@ are even zero divisors and x;.x; # 0 for all i,j = 1,---,n. Notice that f,(Z,) = I(Zp) U {0,¢,21,°+: ,2n} is subsemigroup under multiplication modulo p, and since p is odd. Then f.(Z— p) has not any zero divisor. Corollary 4.1 If Z, be semigroup under multiplication modulo p and p = (q)"" ,meéN. Then f:(Zp) = I(Zp) U {0, geo is subsemigroup under multiplication modulo p. Example 3. (1) Let Zio = {0,1,2,3,--- ,9} be semigroup under multiplication modulo 10. Since p = 10 is even then the z.d.f. map defined as follows: 0 ifx=0; f-z)={ 5 ifxe {2,4,8,6}; x ifae {1,3,7,9,5}. Thus f.(Zio) = {0,1,3,7,9,5}. Obviously, f,(Zio) is a subsemigroup under multiplication modulo 10. Since 410, then f,(Zi9) has not any zero divisor. (2) Let Zi2 = {0,1, 2,3, cdots, 11} be a semigroup under multiplication modulo 12. Since p = 12 is even then the z.d.f map defined as follows: 0 ifx=0; fz(z)=¢ 6 if x € {2,4,6,8, 10}; a iffe {1,3,5, 7,9}. Hence f.(Z12) = {0,1,3,7,9,5,6}. Obviously, f.(Zi2) is a subsemigroup under multiplication modulo 12 and 6 is a zero divisor. 58 Mohamed Esmail Basher References 1 E.EL-Kholy and M. EL-Ghoul, Simplicial foldings, Journal of the Faculty of Education, 18(1993), 443-455. H.R.Farran, E.EL-Kholy and $.A.Robertson, Folding a surface to a polygon, Geometriae Dedicata, 63(1996), 255-266. John A.Beachy, William D.Blair, Abstract Algebra (3th edition), Waveland Pr. Inc, 2006. J.M.Howie, An Introduction to Semigroup Theory, Academic Press, 1976. M.Basher, On the Folding of Finite Topological Space, International Mathematical Forum, Vol. 7, No.15(2012), 745 - 752. M.EI-Ghoul, M.Basher, The invariant of immersions under isotwist folding, KYUNGPOOK Math. J., 46(2006), 139-144. M.El-Ghoul, $.I.Nada and R.M.Abo Elanin, On the folding of rings, International Journal of Algebra, 3(2009), 475-482. M.El-Ghoul , Folding of Manifolds, Ph.D. Thesis, Tanta Univ., Egypt, 1985. P.DL-Francessco, Folding and coloring problem in mathematics and physics, Bulletin of the American Mathematics Socitey, 37(2000), 251-307. 5.A.Robertson, Isometric folding of Riemannian manifolds, Proceedings of the Royal Society of Edinburgh, 79:3-4(1977), 275-284. Math.Combin. Book Ser. Vol.2(2012), 59-70 On Set-Semigraceful Graphs Ullas Thomas Department of Basic Sciences, Amal Jyothi College of Engineering Koovappally P.O. - 686 518, Kottayam, Kerala, India Sunil C. Mathew Department of Mathematics, St.Thomas College Palai, Arunapuram P.O. - 686 574, Kottayam, Kerala, India E-mail: ullasmanickathu@rediffmail.com, sunilcmathew@gmail.com Abstract: This paper studies certain properties of set-semigraceful graphs and obtain certain bounds for the order and size of such graphs. More set-semigraceful graphs from given ones are also obtained through various graph theoretic methods. Key Words: Set-indexer, set-graceful, set-semigraceful. AMS(2010): 05C78 §1. Introduction In 1986, B. D. Acharya introduced the concept of set-indexer of a graph G which is an as- signment of distinct subsets of a finite set X to the vertices of G subject to certain conditions. Based on this, the notions of set-graceful and set-semigraceful graphs were derived. Later many authors have studied about set-graceful graphs and obtained many significant results. This paper sheds more light on set-semigraceful graphs. Apart from many classes of set- semigraceful graphs, several properties of them are also investigated. Certain bounds for the order and size of set-semigraceful graphs are derived. More set-semigraceful graphs from given ones are also obtained through various techniques of graph theory. §2. Preliminaries In this section we include certain definitions and known results needed for the subsequent development of the study. Throughout this paper, J, m and n stand for natural numbers without restrictions unless and otherwise mentioned. For a nonempty set X, the set of all subsets of X is denoted by 2*. We always denote a graph under consideration by G and its vertex and edge sets by V and EF respectively and G’ being a subgraph of a graph G is denoted by G’ C G. When G’ is a proper subgraph of G we denote it by G’ C G. By the term graph we mean a simple graph and the basic notations and definitions of graph theory are assumed to be familiar to the readers. 1Received December 1, 2011. Accepted June 15, 2012. 60 Ullas Thomas and Sunil C. Mathew Definition 2.1({1]) Let G = (V, E) be a given graph and X be a nonempty set. Then a mapping f:V 2%, orf: E> 2%, orf: VUE — 2% is called a set-assignment or set-valuation of the vertices or edges or both. Definition 2.2([1]) Let G be a given graph and X be a nonempty set. Then a set-valuation f: VUE — 2% is a set-indexer of G if 1. f(uv) = f(u) @ f(v),Vuv © E, where ‘®’ denotes the binary operation of taking the symmetric difference of the sets in 2* 2. the restriction maps f\|v and f\z are both injective. In this case, X is called an indexing set of G. Clearly a graph can have many indexing sets and the minimum of the cardinalities of the indexing sets is said to be the set-indexing number of G, denoted by 7(G). The set-indexing number of trivial graph K, is defined to be zero. Theorem 2.3([1]) Every graph has a set-indezer. Theorem 2.4([1]) Jf X is an indexing set of G=(V,E). Then (i) |E| <2'*!—1 and (it) [log.(|E| + 1)] < y(G) < |V| —1, where | ] is the ceiling function. Theorem 2.5({1]) If G’ is subgraph of G, then 7(G’) < 7(G). Theorem 2.6([3]) The set-indexing number of the Heawood Graph is 5. Theorem 2.7([2]) The set-indexing number of the Peterson graph is 5. Theorem 2.8([13]) Jf G is a graph of order n, then y(G) > y(Kijn-1). Theorem 2.9((13]) (Kim) =n +1 if and only if 2" <m < 2"+1—-1, Theorem 2.10([15]) y(Pm) =n +1, where 2" <m< 2"+1-1, Definition 2.11([17]) The double star graph ST(m,n) is the graph formed by two stars Kim and Ky, by joining their centers by an edge. Theorem 2.12((16]) For a double star graph ST(m,n) with |V| = 2!, 1 > 2, l if m is even, y(ST(m,n)) = eat [+1 ifm is odd. Theorem 2.13((16]) y(ST(m,n)) =14+1 if2'+1<|V| < 24 -1;1>2. Theorem 2.14([1]) y(Cs) = 4. Theorem 2.15([1]) y(Ce) = 4. On Set-Semigraceful Graphs 61 Definition 2.16({11]) The join Ky V Py-1 of Ky and P,_ is called a fan graph and is denoted by Fy. Theorem 2.17([15]) y(Fn) =m+1, where m = [logon] and n> 4. Definition 2.18([6]) The double fan graph is obtained by joining P, and Ko. n-1 fl<n<5 n-2 f6<n<7 Theorem 2.19([1]) y(Kn) = 6 if 8<n<9 Theorem 2.20((1]) y(n) =47 if 10<n< 12 8 if B<n<15 Definition 2.21({10]) For a graph G, the splitting graph S'(G) is obtained from G by adding for each vertex v of G, a new vertex say v' so that v' is adjacent to every vertex that is adjacent to v. Definition 2.22([4]) An n-sun is a graph that consists of a cycle C,;, and an edge terminating in a vertex of degree one attached to each vertex of Cy. Definition 2.23((8]) The wheel graph with n spokes, W,,, is the graph that consists of an n-cycle and one additional vertex, say u, that 1s adjacent to all the vertices of the cycle. Definition 2.24([11]) The helm graph H,, is the graph obtained from a wheel W, = Cy, V Ky by attaching a pendant edge at each verter of Cy. Definition 2.25({10]) The twing is a graph obtained from a path by attaching exactly two pendant edges to each internal vertex of the path. Definition 2.26((3]) The triangular book is the graph KyV Nm, where Ny is the null graph of order m. Definition 2.27((6]) The Gear graph is obtained from the wheel by adding a vertex between every pair of adjacent vertices of the cycle. Definition 2.28({10]) Embedding is a mapping ¢ of the vertices of G into the set of vertices of a graph H such that the subgraph induced by the set {¢(u) : u€ V(G)} is isomorphic to G; for all practical purposes, we shall assume then that G is indeed a subgraph of H. Definition 2.29([1]) A graph G is said to be set-graceful if 7(G) = logo(|E| +1) and the corresponding optimal set-indexer is called a set-graceful labeling of G. Theorem 2.30([9]) Every cycle Con_1, n > 2 is set-graceful. Theorem 2.31([10]) K3,5 is not set-graceful. Theorem 2.32([15]) y(Han_1) =n+4+2 for n> 2. 62 Ullas Thomas and Sunil C. Mathew Theorem 2.33([1]) The star Ky2n_1 is set-graceful. §3. Certain Properties of Set-Semigraceful Graphs In this section we derive certain properties of set-semigraceful graphs and obtain certain bounds for the order and size of such graphs. Definition 3.1([1]) A graph G is said to be set-semigraceful if y(G) = [log2(|E| + 1)]. Remark 3.2 (1) The Heawood graph is set-semigraceful by Theorem 2.6. (2) The stars are set-semigraceful by Theorem 2.9. (3) The Paths P,; n 4 2™, m > 2 are set-semigraceful by Theorem 2.10. (4) Ky, is set-semigraceful if n € {1,--- ,7,9,12} by Theorems 2.19 and 2.20. (5) The Double Stars ST(m,n); m+n 4 2!, m is odd are set-semigraceful by Theorems 2.12 and 2.13. (6) The helm graph H2n_ is set-semigraceful by Theorem 2.32. (7) All set graceful graphs are set-semigraceful. (8) The path P; is set-semigraceful but it is not set-graceful. (9) A graph G of size 2” — 1 is set-semigraceful if and only if it is set-graceful. (10) Not all graphs of size 2” — 1 is set-semigraceful. For example K3,5 is not set- semigraceful by Theorem 2.31. The following theorem is an immediate consequence of the above definition. Theorem 3.3 Let G be a (p,q)-graph with y(G) =m. Then G is set-semigraceful if and only if 27-1 <q@<2™-1. Corollary 3.4 Let G’ be a (p,q') spanning subgraph of a set-semigraceful (p,q)-graph G with qd >2-!, Then G' is set-semigraceful. Proof The proof follows from Theorems 2.4 and 3.3. Remark 3.5 (1) The Peterson graph is not set-semigraceful by Theorem 2.7. (2) The paths Pym; m > 2 are not set-semigraceful by Theorem 2.10. (3) The double stars ST (m,n); m+n = 2!, m is odd are not set-semigraceful by Theorem 2.12. (4) K,, is not set-semigraceful if n € {8,10, 11,13, 14,15} by Theorem 2.20. Corollary 3.6 Let T be a set-semigraceful tree of order p, then 21-141 < p< 21, While Theorem 3.3 gives bounds for the size of a set-semigraceful graph in terms of the set-indexing number, the following one gives the same in terms of its order. Theorem 3.7 Let G be a set-semigraceful (p,q)-graph. Then gllogap|—1 Sas 9 | loge (PL +1)] 24. On Set-Semigraceful Graphs 63 Proof By Theorems 2.4, 2.5 and 2.8, we have logan] <1 Kaye) $(G) = [logela +191 < tom Z2 2 +a). Now by Theorem 3.3 we have gllogep|—1 <q< 9 | loge (PL +1)] a Remark 3.8 The converse of Theorem 3.7 is not always true. By Theorem 2.14 we have, [loga(|E(Cs)| + 1)] = [log26] = 3 < (Cs) = 4. But Cs is not set-semigraceful even if 2? < |E(Cs)| < 24—1 holds. Further as a consequence of the above theorem we have the graphs Ce U3Ky, C5 U4Kky and Cs U 2K are not set-semigraceful. Remark 3.9 By Theorem 2.5, for any subgraph G’ of G, y(G’) < 7(G). But subgraphs of a set-semigraceful graph need not be set-semigraceful. For example Kg is set-semigraceful but the spanning subgraph Cg of Kg is not set-semigraceful, by Theorem 2.19. In fact the result given by Theorem 3.3 holds for any set-semigraceful graph as we see in the following. Theorem 3.10 Every connected set-semigraceful (p,p — 1)-graph is a tree such that gm-1 4 1 Ke D < gm and for every m, such a tree always exists. Proof Clearly every connected (p,p— 1) graph T is a tree and by Theorem 3.3 we have 2™-l11<p< 2” if T is set-semigraceful. On the other hand, for a given m, the star graph Kj ,2m_ 1 is set-semigraceful. Theorem 3.11 Jf the complete graph Ky; n > 2 is set-semigraceful then 2m—1<7(Kn) <2m+1, where, m= |logon|. Proof If Ky is set-semigraceful then fio +1] = UK») > [tom | [logon + loge(n — 1) — loge2] I I [logon + logg(n —1)— 1]. For any n, there exists m such that 2™ <n < 2™*!— 1 so that from above y(K,) > 2m —1. But we have n(n —1)+2 1) = flon®&=2 44] = fia 222] om 64 Ullas Thomas and Sunil C. Mathew Thus we have 2m — 1 < 7(Kpn) < 2m+1; m= |logan|). Remark 3.12 The converse of Theorem 3.11 is not always true. For example, by Theorem 2.20 we have 2 |logon| — 1 < y(Kg) < 2 |logan| +1. But the complete graph Kg is not set-semigraceful. Also by Theorems 2.20 and 3.11 we have ky3, Ky4 and Ky5 are not set-semigraceful. Theorem 3.13 If a (p,q)-graph G has a set-semigraceful labeling with respect to a set X of cardinality m > 2, there exists a partition of the vertex set V(G) into two nonempty sets V, and V2 such that the number of edges joining the vertices of Vi with those of V2 is at most 2™—!. Proof Let f : VUE — 2* be a set-semigraceful labeling of G with indexing set X of cardinality m. Let Vi = {u € V : |f(u)| is odd} and V2 = V—Vy. We have |A @ B| = |A| + |B| — 2|A/N B| for any two subsets A, B of X and hence |A @ B| = 1(mod 2) > A and B does not belong to the same set Vj; 7 = 1,2. Therefore all odd cardinality subsets of X in f(£) must appear on edges joining V; and V2. Consequently there exists at most 2”~! edges between V; and V3. Remark 3.14 In 1986, B.D.Acharya [1] conjectured that the cycle Cgn_1; n > 2 is set-graceful and in 1989, Mollard and Payan [9] settled this in the affirmative. The idea of their proof is the following: Consider the field GF(2”) constructed by a binary primitive polynomial say p(a) of degree n. Let a be a root of p(x) in GF(2"). Then GF(2") = {0,1,a,a7,...,a2"~?}. Now by assigning a’~!mod p(a), 1 < i < 2"—1, to the vertices v; of the cycle Cgn_1 = (v1,...,Van_1, 1) we get a set-graceful labeling of Cyn_1 with indexing set X = {1,a,a?,...,a@”~!}. Note that here aJmod p(a) = apa® + ayat +...+@n_10"1; a; = 0 or 1 for 0 <i <n-1 with a® =1 and we identify it as {aja’/ a; = 1; 0 <i<n-—1} which is a subset of X. Theorem 3.15 All cycles Cy with 2" -1<k <2"+2"-!—2,n>3 are set-semigraceful. Proof The cycle Cgn_1 = (v1, v2,°++ ,Von—1, 01) has a set-graceful labeling f as described in the above Remark 3.14. Take 1 = 2”~! — 1 new vertices u1,---,u,; and form the cycle Conqi-1 = (U1, U1, V2, U3, U2, V4, US, UZ, U6, U7, °° * » Von—3, Ul, Van—2, Von-1, U1). Now define a set- indexer g of Cgn4,-1 with indexing set Y = X U {x} as follows: g(u;) = f(vi); 1 <i < 2"-1 and g(u;) = f(vaj-1, v2;) U {x}; 1 < 7 <1. Then by Theorem 2.4 we have y(C2n4;-1) =n +1. Now by removing the vertices u;; 2 < 7 < 1 and joining v2;_1v2; in succession we get the cycles Con41~-2, Con41-3,...,Con. Clearly g induces optimal set-indexers for these cycles by Theorem 2.4 and 7(C,) =n +1; 2% <k < 2”+1-—2 so that these cycles are set-semigraceful. Theorem 3.16 The set-indexing number of the twing graph obtained from Pan_1 isn+2;n > 3 and hence it is set-semigraceful. Proof Let Pan_1 = (v1,...,Van-1). By Theorem 2.10 we have y(Pon_1) =n. Let f be On Set-Semigraceful Graphs 65 an optimal set-indexer of Pan_1 with indexing set X. Let T be a twing graph obtained from Pyn_1 by joining each vertex v;; i € {2,3,---,2” — 2} of Pon_, to two new vertices say u; and w; by pendant edges. Consider the set-indexer g of T with indexing set Y = X U {x, y} defined as follows: g(v) = f(v) for all v © V(Pan_1), g(us) = f(wi-1) U {x} and g(w;) = f(ui-1) U {y}; 2<i< 2"—2. Consequently 7(T) <n-+2. But by Theorem 2.4 we have [loge(|E(T)| + 1)] = floge(2” —2+2"-342"-341)]) =n4+2< (1). Thus T is set-semigraceful. §4. Construction of Set-Semigraceful Graphs In this section we construct more set-semigraceful graphs from given ones through various graph theoretic methods. Theorem 4.1 Every set-semigraceful (p,q)-graph G with 7(G) = m can be embedded in a set-semigraceful (2™, q)-graph. Proof Let f be a set-semigraceful labeling of G with indexing set X of cardinality m = ¥(G). Now add 2™ — p isolated vertices to G and assign the unassigned subsets of X under f to these vertices in a one to one manner. Clearly the resulting graph is set-semigraceful. Theorem 4.2 A graph G is set-semigraceful with y(G) =m, then every subgraph H of G with 2™-1 < |E(H)| < 2™—1 is also set-semigraceful. Proof Since every set-indexer of G is a set-indexer of H, the result follows from Theorem 2.4. Corollary 4.3 All subgraphs G of the star Ky, is set-semigraceful with the same set-indexing number m if and only if 2™-* < |E(G)| < 2™—-1. Proof The proof follows from Theorems 4.2 and 2.9. Theorem 4.4 Jf a (p,p—1)-graph G is set-semigraceful, then GV Non_, is set-semigraceful. Proof Let G be set-semigraceful with set-indexing number m. By Theorem 2.4 we have V(GV Noni) 2 [loge(|E(GV Non-1)| + 1)] [logo(p —1+ p(2”— 1) +1)] [logop(2”)] = [loge (2”) + loga(p)| = n+m. I l| Let f be a set-semigraceful labeling of G with indexing set X of cardinality m. Consider the set Y = {yi,.--,Yn} and let V(Non_1) = {v1,..., Un}. We can find a set-semigraceful labeling say g of GV Non_1 with indexing set X UY as follows: g(u) = f(u) for all u € V(G) and assign the distinct nonempty subsets of Y to the vertices v1,...,Un in any order. Thus GV Non_, is set-semigraceful. 66 Ullas Thomas and Sunil C. Mathew Remark 4.5 The converse of Theorem 4.4 is not true in general. For example consider the wheel graph Ws = Cs V Ky = (ui, ue,...Us, ur) V {u}. Now assigning the subsets {a}, {a, bd}, {a,b,c}, {a,d}, {a,b,c,d} and ¢ of the set X = {a,b,c,d} to the vertices u1,...,us and u in that order we get Ws as set-semigraceful whereas C's is not set-semigraceful by Theorem 2.14. Corollary 4.6 The triangular book Kz V Ngn_, is set-semigraceful. Proof The proof follows from Theorems 4.4 and 2.10. Theorem 4.7 The fan graph F,, is set-semigraceful if and only ifn A 2™+1; n> 4. Proof Ifn—1 #2, by Theorem 2.10 we have P,,_1 is set-semigraceful so that by Theorem 4.4, F,, = Pr_1 V Ky is set-semigraceful. Conversely if F;, is set-semigraceful, then by Theorem 3.3, we have V(Fr) = [loge(n —1+n—2+4+1)] = [loge(2n — 2)] = [loge(n —1)] +1. But by Theorem 2.17 we already have y(F;,) = [logon] + 1. Consequently we must have nA~2™ +1. Theorem 4.8 Every graph can be embedded as an induced subgraph of a connected set- semigraceful graph. Proof Let {v1,...,Un} be the vertex set of the given graph G. Now take a new vertex say u and join it with all the vertices of G. Consider the set X = {a1,...,a%,}. Let m = 2” — (|E| +n) —1. Take m new vertices u4,...,tm and join them with wu. A set-indexer of the resulting graph G’ can be defined as follows: Assign ¢ to u and {x;} to vj; 1 <i <n. Let S={f(e): e€ E}U {{ai}: 1<i<n}. Note that |S| = |E£|+ 7. Now by assigning the m elements of 2* — (SU @) to the vertices u1,...,tm in any order we get a set-indexer of G’ with X as the indexing set, making G’ set-semigraceful. Theorem 4.9 The splitting graph S'(G) of a set-semigraceful bipartite (p,q)-graph G with 7(G) =m and 3q > 2™*"1, is set-semigraceful. Proof Let f be an optimal set-indexer of G with indexing set X of cardinality m. Let Vi = {v1,...,Un} and V2 = {u1,...,uz} be the partition of V(G), where n = p—k. Since G is set-semigraceful with y(G) =m, by Theorem 3.3 we have 2™ 1! < q < 2™—1. To form the / J joining v; or uj, to all neighbours of v; or u; in G respectively. Since S’(G) has 3q edges, by splitting graph S’(G) of G, for each v; or u; in G, add a new vertex v; or wu’, and add edges Theorem 2.4 we have 7($'(G)) = [loga(|E(S'(G))| + 1)] = [loga(3q + 1)] = [loga(2""* + 1)] =m+2. We can define a set-indexer g of S’(G) with indexing set Y = X U {x,y} as follows: g(v) = f(v) for all v € V(G), g(v;) = flv) U {x}; 1 < i <n and g(uj) = f(us)U{y}s 1 <7 < L. Consequently +(S'(G)) =m +2 = [logs(|E($'(G))| + 1 On Set-Semigraceful Graphs 67 and hence $’(G) is set-semigraceful. Remark 4.10 (1) Even though C3 is not bipartite, both C3 and its splitting graph are set- semigraceful. (2) Splitting graph of a path P, is set-semigraceful. But P, is not set-semigraceful. Theorem 4.11 For any set-graceful graph G, the graph H; GU Ki CH CGV Ky, is set- semigraceful. Proof Let m = 7(G) = loga(|E(G)| + 1). Then by Theorem 2.4 we have YH) 2 [loge(|E(H)| + 1)] = [loga(2™ + 1)] = m+). Let f be a set-graceful labeling of G with indexing set X. Now we can extend f to a set-indexer g of GV Ky with indexing set Y = X U {x} of cardinality m+ 1 as follows: g(u) = f(u) for all u € V(G) and g(v) = {x} where {v} = V(G). Clearly g(e) = f(e) for all e € E(G) and g(uv) = g(u) U {a} are all distinct. Then by Theorem 2.5 we have (A) =m+1= [logs(|E(A)| +1). Corollary 4.12 The wheel Wan_, is set-semigraceful. Proof The proof follows from Theorems 4.11 and 2.30. Theorem 4.13 Let G be a set-graceful (p,p — 1)-graph, then GV Ny», is set-semigraceful. Proof Let G be set-graceful graph with set-indexing number n. For every m, there exists I such that 2! < m < 2'+1— 1. By Theorem 2.4 we have VGV Nm) 2 [log2(|E(GV Nm)| +1)] = [loge(p—1+ pm + 1)] = [logap(m + 1)] = [log2(2”)(m + 1)] = [log2(2”) + logo(m+1)] >n+ 141. Let f be a set-semigraceful labeling of G with X; |X| = n as the indexing set. Consider the set Y = {y,---, yi} and V(Nin) = {v1,...,Um}. Now we can extend f to GV N,, by assigning the distinct nonempty subsets of Y to the vertices v1, ..., Um in that order to get a set-indexer of GV N,, with indexing set X UY. Hence GV N,,, is set-semigraceful. Corollary 4.14 Ky,2n_1,m is set-semigraceful. Proof The proof follows from Theorems 2.33 and 4.13. Theorem 4.15 Let G be a (p,p — 1) set-graceful graph, then GV Ky and GV Kz are set- semigraceful. Proof Let f be a set-graceful labeling of G with indexing set X of cardinality n. By 68 Ullas Thomas and Sunil C. Mathew Theorem 2.4 we have IV [loge (|E (GV K2)| + 1)] = [loge(p —1+2p+1+4+1)] 1 = [logo(3p+1)] = tos + loge(p + >| 1 = logs + logg(2” + >| >n+2. Consider the set Y = X U {&n41,%n+2} and the set-indexer g of GV K2 defined by g(u) = f(u) for all u € V(G), g(u1) = {an41} and g(v2) = {an+2}; v1, ve € V(K2). Consequently (GV Ko) =n+1 so that it is set-semigraceful. Let K3 = (v1, v2, 03,01). By Theorem 2.4 we have 7(G V K3) > [loga(|E(G V K3)| + 1)] =[loge(p —1+3p+3+41)] = [loge(4p + 3)] = [logo(4.2” + 3)] = [logo(2"t? +3)| > n+3. We can find a set-indexer h of GV Ks with indexing set Z = Y U {2,+3} as follows: Assign {tn+1}, {@n42} and {r,+3} to the vertices of K3 and h(u) = f(u) for all ue V(G). Clearly h is a set-semigraceful labeling of GV K3. Corollary 4.16 All double fans Py V Ko;n 4 2™, m> 2 are set-semigraceful. Proof The proof follows from Theorems 4.15 and 2.10. Corollary 4.17 The graph Ky,9n_1 V Kg ts set-semigraceful. Proof The proof follows from Theorems 4.15 and 2.13. Theorem 4.18 If C;, is set-semigraceful, then the graph C,,V K2 is set-semigraceful. Moreover (Cn V Ko) =m+ 2, where 2™<n<2™+2™1_2 n>7. Proof The proof follows from Theorems 3.15 and 4.15. Theorem 4.19 Let G be a set-semigraceful (p,q)-graph with y(G) =m. If p > 2™ 1, then GV ky set-semigraceful. Proof By Theorem 3.3 we have 2™~! < |E(G)| < 2™ — 1. Since |V| > 2™~1, by Theorem 2.4 we have (GV Ki) > [logo(|E| +1)] = m+1. Let f be a set-indexer of G with indexing set X of cardinality m = 7(G). Now we can define a set-indexer g of GV Ky; V(k1) = {vu} with indexing set Y = X U {2} as follows: g(u) = f(u) for all u € V(G) and g(v) = Y. This shows that GV K; is set-semigraceful. Corollary 4.20 If Cn is set-semigraceful, then Wm 1s also set-semigraceful. Proof The proof follows from Theorem 4.19. Corollary 4.21 W,, is set-semigraceful, where 2™+1<n<2™+2™-1_ 1], proof The proof follows from Theorem 3.15 and Corollary 4.20. On Set-Semigraceful Graphs 69 Theorem 4.22 The gear graph of order 2n+1 with 2°-1<n< gm-l4gm-3) m > 3 is set-semigraceful. Proof The proof follows from Theorem 2.5 and Corollary 4.21. Theorem 4.23 Let G be a set-semigraceful hamiltonian (p,q)-graph with y(G) = m and p> 27-1, IfG’ is a graph obtained from G by joining a pendand vertex to each vertex of G, then G’ is set-semigraceful. Proof Let C = (v1, v2,-++,Un,v1) be a hamiltonian cycle in G. Let f be a set-indexer of G with 7(G) = m and X be the corresponding indexing set. Now take n new vertices vy 1 <i < mand let G’ = GU {ujyu;/ 1 < i < n}. By Theorem 2.4 we have 7(G’) > [log2(|E(G’)| + 1)] = m-+1. We can define a set-indexer g of G’ with indexing set Y = XU {x} as follows: g(u) = f(u) for all u € V(G), g(vi) = f(uivigr) U {x}; 1 <a <n with vnqi = v1. Clearly G’ is set-semigraceful. Corollary 4.24 IfC,, is set-semigraceful, then the sun-graph obtained from C, is set-semigraceful. Proof The proof follows from Theorem 4.23. Corollary 4.25 The sun-graph of order 2n;2™ <n <2™+4+2™—-1_2:m > 3 is set-semigraceful. Proof The proof follows from Theorem 3.15 and Corollary 4.24. References [1] B.D.Acharya, Set valuations of graphs and their applications, Proc. Sympos. on Optimiza- tion, Design of Experiments and Graph Theory, Indian Institute of Technology Bombay, 1986, 231-238. [2] B.D.Acharya, Set indexers of a graph and set graceful graphs, Bull. Allahabad Math. Soc., 16 (2001), 1-23. [3] B.D.Acharya, S.Arumugam and A.Rosa, Labelings of Discrete Structures and Applications, Narosa Publishing House, New Delhi, 2008. [4] R.Anitha and R.S.Lekshmi, N-sun decomposition of complete, complete-bipartite and some Harary graphs, International Journal of Computational and Mathematical Sciences, 2(2008), 33-38. 5] G.Chartrand and P.Zhang, Introduction to Graph Theory, Tata Mcgraw Hill, New Delhi, 2005. 6| J.A.Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 13 (2010). 7| S.M.Hegde, On set valuations of graphs, Nat. Acad. Sci. Letteres, 14 (1991), 181-182. 8] A.Kirlangic, The repture degree and gear graphs, Bull. Malays. Math. Sci. Soc., 32 (2009), 31-36. 9] M.Mollard and C.Payan, On two conjectures about set-graceful graphs, European J. Com- 70 10 11 12 13 14 15 16 17 Ullas Thomas and Sunil C. Mathew binatorics 10 (1989), 185-187. K.L.Princy, Some Studies on Set Valuations of Graphs-Embedding and NP Completeness, Ph.D. Thesis, Kannur University, 2007. W.C.Shiu and P.C.B.Lam, Super-edge-graceful labelings of multi-level wheel graphs, fan graphs and actinia graphs, Congr. Numerantium, 174 (2005), 49-63. U.Thomas and S.C.Mathew, On set indexers of multi-partite graphs, STARS: Int. Journal (Sciences), 3 (2009), 1-10. U.Thomas and $.C.Mathew, On set indexer number of complete k-partite graphs, Int. J. Math. Computation, 5 (2009), 18-28. U.Thomas and S.C.Mathew, On topological set indexers of graphs, Advances and Applica- tions in Discrete Mathematics, 5 (2010), 115-130. U. Thomas and S. C. Mathew, On set indexers of paths, cycles and certain related graphs, Discrete Mathematics, Algorithms and Applications (Accepted). U.Thomas and $.C.Mathew, Topologically set graceful stars, paths and related graphs, South Asian Journal of Mathematics (Accepted). T.M.Wang and C.C.Hsiao, New constructions of antimagic graph labeling, Proc. 24th Workshop on Combinatorial Mathematics and Computation Theory, National Chi Nan University Taiwan, 2007, 267-272. Math.Combin. Book Ser. Vol.2(2012), 71-75 On Generalized m-Power Matrices and Transformations Suhua Ye!?, Yizhi Chen! and Hui Luo! 1. Department of Mathematics, Huizhou University, Huizhou 516007, P.R.China 2. School of Mathematical Science, South China Normal University, Guangzhou 510631, P.R.China E-mail: yizhichen1980@126.com Abstract: In this paper, generalized m—power matrices and generalized m—power trans- formations are defined and studied. First, we give two equivalent characterizations of gen- eralized m—power matrices, and extend the corresponding results about m— idempotent matrices and m—unit-ponent matrices. And then, we also generalize the relative results of generalized m—power matrices to the ones of generalized m—power transformations. Key Words: Generalized m—power matrix, generalized m—power transformation, equiv- alent characterization. AMS(2010): 15A24 §1. Introduction The m-idempotent matrices and m—unit-ponent matrices are two typical matrices and have many interesting properties (for example, see [1]-[5]). A matrix A € C”*” is called an m—idempotent (m—unit-ponent) matrix if there exists positive integer m such that A™ = A(A™ = I). Notice that A™ = A if and only if [[(A+eil) =O, i=l where €; = 0, €2,€3,°-* ,Em are the m— 1 power unit roots, A™ = 1 if and only if [[(A+eil) =O, i=1 where €1, €2,°°*,€m are the m—power unit roots. Naturally, we will consider the class of matrices which satisfies that m [[4+4) =o, i=1 where \j, A2,°-: ,Am are the pairwise different complex numbers. lSupported by Grants of National Natural Science Foundation of China (No.11101174); Natural Science Foundation of Guangdong Province (No.S2011040003984); Natural Science Foundation of Huizhou University (C211 - 0106); Key Discipline Foundation of Huizhou University. 2Received February 27, 2012. Accepted June 16, 2012. 72 Suhua Ye, Yizhi Chen and Hui Luo For convenience, we call a matrix A € C”*” to be a generalized m—power matrix if it satisfies that [Jj ,(A + AZ) = O, where 1, A2,--- ,Am are the pairwise different complex numbers. In this paper, we firstly study the generalized m—power matrices, and give two equiv- alent characterizations of such matrices. Consequently, the corresponding results about m— idempotent matrices and m—unit-ponent matrices are generalized. And then, we also define the generalized m—power transformations, and generalize the relative results of generalized m-—power matrices to those of generalized m—power transformations. For terminologies and notations occurred but not mentioned in this paper, the readers are referred to the reference [6]. §2. Generalized m—Power Matrices In this section, we are going to study some equivalent characterizations of generalized m—power matrices. First, we introduce some lemmas following. Lemma 2.1([4]) Let A1,A2,--: ,Am be the pairwise different complez numbers and A € C"*". Then - a r([[(A+A2)) = So r(A+ AD) - (m= 1)n. i=l i=1 Lemma 2.2((4]) Let fi(x), fa(x),--+,fm(x) € Cla] be pairwisely co-prime and A € C™*". Then m Yo r(f(A)) = (m= 1)n-4 eFC). Lemma 2.3([1]) Assume that f(x), g(a) € C[x], d(x) = (f(x), g(x)) and m(A) = [f (x), g(a)]. Then for any AE C"*", (f(A) + r(g(A)) = r(d(A)) + r(m(A)). Theorem 2.4 Let X41, 2,°°: ,Am € C be the pairwise different complex numbers and A € C"*". Then |], (A+ AL) = O if and only if i r(A+ AL) = (m= 1)n. Proof Assume that JT}, (A+A:iJ) = O, by Lemma 2.1, we can immediately get 07", r(A+ Ail) = (m— 1)n. Assume that )7/", r(A + AZ) = (m—1)n. Take fi(z) = «+ Ai(i = 1,2,--- ,m), where Ni FA; if 4 AJ. Clearly, we have (f;(x), f;(a)) =1ifi #7. Now, by Lemma 2.2, m m do r(Fi(A)) = (m — 1m + r(T] (F(A). i=l i=l Also, since >)", r(A + AL) = (m— 1)n, we can get r([] i, (fi(A))) = 0, this implies that Ue +A,2) =O. i=l Generalized m-Power Matrices and Generalized m-Power Transformations 73 By Theorem 2.4, we can obtain the following conclusions. Consequently, the corresponding result in [3] is generalized. Corollary 2.5 Let €1,€2,--+ ,€m—1 € C be the m—1 power unit roots and A € C"*". Then A™ =A if and only if r(A) + r(A-— e1l) + 7(A — 01) +--+ +r (A = emit) = (m— In. Corollary 2.6 Let €1,€2,:-:,&m € C be the m power unit roots and A € C"*". Then A™ =I if and only if r(A— e1I) + 7r(A — eof) +--+ + 7r(A = eml) = (m—1)n. Now, we give another equivalent characterizations of the generalized m—power matrices. Theorem 2.7 Let X41, 2,-°: ,Am € C be the pairwise different complex numbers and A € C"*", Then 7 (m — 2)(m—1)n [ [44D =0 ff and only if SO r((A+AT)(A+A4D)) = eer vaca i=1 1<i<j<m Proof Assume that [Jj".,(A + Ai) =O. Then r([][j".,(A + AiL)) = 0. Notice that S> r((A+ Al)(A + A5J)) a5 er r((A+AI)(A + A;1)) 1<i<j<m i=1 j=i+1 and by Lemmas 2.2 and 2.3, it is not hard to get that Sor((A+ at)(A +240) = = ADa rq aso) = (m—2)-r(A+A,J), Sor(A+dat\(A jf) +r(At UD) =n+ (m—3)-r(A+ rol), a r((A+A3l)(A + Aj) +r(A+ AD) + r(A + Aol) = 2n + (m—4)-r(A+As]), > T((A + Am—21)(A + AzI)) + r(A+ AI) = (m—3)-n+r(At+Am-_2I), a. i=l m—2 r((A+Am—12)(A + AmL)) + $5 r(A + AL) = (m—2)-n 1=1 Thus, we have y: r((A+AD)(A+A;D) = (m= 2)(m= Vn — 2 1<i<j<m From the discussions above, we have S> r((AtAI)(A+AjD)) = (mtn =n + (m—1)-r(]](A+,2)). 1<i<j<m i=1 74 Suhua Ye, Yizhi Chen and Hui Luo Hence, if —2 -1 S> r((A+ AIA +51) = SSS 1<i<j<m then Ps r([[(4 +X) = 0, i=1 Le., [[44.4) =0. i=1 By Theorem 2.7, we can get corollaries following. Also, the corresponding result in [3] is generalized. Corollary 2.8 Let €1,€2,:+:,€m—1 € C be the m—1 power unit roots and A € C”*". Then A™ = A if and only if r(A(A — e12)) +--+ + r(A(A = Emil) + 7r((A — e12)(A — eof) 4+ + r((A—e11)(A— em—11)) +--+ r((A — Em—21)(A — €m—11)) = mn Corollary 2.9 Let €,€2,---,ém € C be the m power unit roots and AE C"*". Then A" =I if and only if ae r((A+eJ)(A+e,1)) = fue 1<i<j<m §3. Generalized m—Power Transformations In this section, analogous with the discussions of the generalized m—power matrices, we will firstly introduce the concepts of generalized m—power linear transformations, and then study some of their properties. Let V be a n dimensional vector space over a field F' and o a linear transformation on V. We call o to be a generalized m—power transformation if it satisfies that m IIe + Vie) =@ i=1 for pairwise different complex numbers j, A2,:-: ,Am, where € is the identical transformation and @ is the null transformation. Especially, o is called an m—idempotent ( m—unit-ponent) transformation if it satisfies that 0™ = o(o™ =). From [6], it is known that n dimensional vector space V over a field F’ is isomorphic to F” and the linear transformation space L(V) is isomorphic to F”*". Thus, we can obtain the following results about generalized m—power transformations whose proofs are similar with the corresponding ones in Section 2. And we omit them here. Theorem 3.1 Let V be an dimensional vector space over a field F and o a linear transfor- mation on V. Then o is a generalized m—power transformation if and only if 3 dimIm(o + Axe) = (mM — 1)n. i=l Generalized m-Power Matrices and Generalized m-Power Transformations 75 By Theorem 3.1, we obtain the following conclusions. Corollary 3.2 Let V be an dimensional vector space over a field F and o a linear transfor- mation on V. Then o is an m— idempotent transformation if and only if dimIm(A) + dimIm(A — 611) + dimIm(A — e921) +--+ + dimIm(A — €m_il) = (m — I)n. Corollary 3.3 Let V be an dimensional vector space over a field F and o a linear transfor- mation on V. Then o is an m—unit-ponent transformation if and only if dimIm(A — e,1) + dimIm(A — eof) +--- + dimIm(A — émI) = (m—1)n. Corollary 4.4 Let V be an dimensional vector space over a field F and o a linear transfor- mation on V. Then o is a generalized m—power transformation if and only if (m — 2)(m—1)n S- dimIm((a + Ax€)(o + Aje)) = ; 1<i<j<m Corollary 3.5 Let V be an dimensional vector space over a field F and o a linear transfor- mation on V. Then o is an m— idempotent transformation if and only if dimIm(a(o — €1€)) +--+ + dimIm(o(o — €m_1€)) +dimIm(o — €1€)(o — €2€)) +--+ dimIm((o — e1€)(o — Em_1€)) —2 -1 +-+++4+dimIm((o — €m—2€)(o — Em_-1€)) = —— Corollary 3.6 Let V be an dimensional vector space over a field F and o a linear transfor- mation on V. Then o is an m—unit-ponent transformation if and only if (m — 2)(m—1)n S- dimIm((o + ex€)(o + €5€)) = : 1<i<j<m References 1] G.Q.Lin, Z.P.Yang, M.X.Chen, A rank identity of matrix polynomial and its application, Journal of Bethua University (Natural science), 9(1)(2008),5-8. 2| H.B.Qiu, Rank identities of a class of matrices, Journal of Guangdong University of Tech- nology, 24(1)(2007), 82-84. 3] J.G.Tang, Q.Y.Yan, Necessary and sufficient conditions for unit-ponent matrices, Mathe- matics in Practice and Theory, 40(20)(2009),172-176. 4) T.T.Wang, B.T.Li, Proof of a class of matrix rank identities, Journal of Shandong Univer- sity (Natural Science), 42(2)(2007), 43-45. 5] Z.P.Yang, Z.X.Lin, An identity of the rank about the idempotent matrix and its application, Journal of Beihua University (Natural science) ,23(2)(2007),141-146. 6] H.R.Zhang and B.X.Hao, Higher Algebra (In Chinese, the 5th edition), Higher Education Press, Beijing, 1999. Math.Combin. Book Ser. Vol.2(2012), 76-80 Perfect Domination Excellent Trees Sharada B. (Department of Studies in Computer Science, University of Mysore, Manasagangothri, Mysore-570006, India) E-mail: sharadab21@gmail.com Abstract: A set D of vertices of a graph G is a perfect dominating set if every vertex in V \ D is adjacent to exactly one vertex in D. In this paper we introduce the concept of perfect domination excellent graph as a graph in which every vertex belongs to some perfect dominating set of minimum cardinality. We also provide a constructive characterization of perfect domination excellent trees. Key Words: Tree, perfect domination, Smarandachely k-dominating set, Smarandachely k-domination number. AMS(2010): 05C69 §1. Introduction Let G = (V, EF) be a graph. A set D of vertices is a perfect dominating set if every vertex in V \ D is adjacent to exactly one vertex in D. The perfect domination number of G, denoted Yp(G), is the minimum cardinality of a perfect dominating set of G. A perfect dominating set of cardinality y,)(G) is called a 7p(G)-set. Generally, a set of vertices S in a graph G is said to be a Smarandachely k-dominating set if each vertex of G is dominated by at least k vertices of S and the Smarandachely k-domination number ¥,(G) of G is the minimum cardinality of a Smarandachely k-dominating set of G. Particularly, if k = 1, such a set is called a dominating set of G and the Smarandachely 1-domination number of G is nothing but the domination number of G and denoted by 7(G). Domination and its parameters are well studied in graph theory. For a survey on this subject one can go through the two books by Haynes et al [3,4]. Sumner [7] defined a graph to be y— excellent if every vertex is in some minimum dominat- ing set. Also, he has characterized y — excellent trees. Similar to this concept, Fricke et al [2] defined a graph to be i — excellent if every vertex is in some minimum independent dominating set. The i-excellent trees have been characterized by Haynes et al [5]. Fricke et al [2] defined yz — excellent graph as a graph in which every vertex is in some minimum total dominating set. The 7-excellent trees have been characterized by Henning [6]. In this paper we introduce the concept of yp-excellent graph. Also, we provide a construc- tive characterization of perfect domination excellent trees. We define the perfect domination number of G relative to the vertex u, denoted 7;/(G), as the minimum cardinality of a perfect dominating set of G that contains u. We call a perfect dominating set of cardinality 7,/(G) containing u to be a ¥(G)-set. We define a graph G to be 1Received February 1, 2012. Accepted June 18, 2012. Perfect Domination Excellent Trees 77 Yp — excellent if 7;;(G) = Yp(G) for every vertex u of G. All graphs considered in this paper are finite and simple. For definitions and notations not given here see [4]. A tree is an acyclic connected graph. A leaf of a tree is a vertex of degree 1. A support vertex is a vertex adjacent to a leaf. A strong support vertex is a support vertex that is adjacent to more than one leaf. §2. Perfect Domination Excellent Graph Proposition 2.1 A path P,, is yp) — excellent if and only ifn = 2 or n= 1(mod3). Proof It is easy to see that the paths P, and P,, for n = 1(mod3) are yp-excellent. Let P,,n > 3, be a yp-excellent path and suppose that n = 0, 2(mod3). If n = 0(mod3), then P,, has a unique Yp-set, which does not include all the vertices. If n = 2(mod3), then no yp-set of P,, contains the third vertex on the path. Proposition 2.2 Every graph is an induced subgraph of a yp)-excellent graph. Proof Consider a graph H and let G = Hok,, the 1-corona of a graph H. Every vertex in V (#) is now a support vertex in G. Therefore, V(#) is a yp-set of G. As well, the set of end vertices in G is a yp-set. Hence every vertex in V(G) is in some 7p-set and G is y,p-excellent. Since H is an induced subgraph of G, the result follows. §3. Characterization of Trees We now provide a constructive characterization of perfect domination excellent trees. We accomplish this by defining a family of labelled trees as defined in [1]. Let F = {Tr }n>1 be the family of trees constructed inductively such that T| is a path Py, and T,, = T, a tree. If n > 2, T;,1 can be obtained recursively from T; by one of the two operations F,, Fz fori =1,2,---,n—1. Then we say that T has length n in F. We define the status of a vertex v, denoted sta(v) to be A or B. Initially if T; = Py, then sta(v) = A if v is a support vertex and sta(v) = B, if v is a leaf. Once a vertex is assigned a status, this status remains unchanged as the tree is constructed. Operation 7, Assume y € T,, and sta(y) = A. The tree T,,41 is obtained from T,, by adding a path x, w and the edge xy. Let sta(x) = A and sta(w) = B. Operation F2 Assume y € T,, and sta(y) = B. The tree T,,41 is obtained from T,, by adding a path x, w,v and the edge xy. Let sta(x) = sta(w) = A and sta(v) = B. F is closed under the two operations F, and Fj. For T € F, let A(T) and B(T) be the sets of vertices of status A and B respectively. We have the following observation, which follow from the construction of F. Observation 3.1 Let T € F and v € V(T) 78 Sharada B. 1. If sta(v) = A, then v is adjacent to exactly one vertex of B(T) and at least one vertex of A(T). 2. If sta(v) = B, then N(v) is a subset of A(T). 3. If v is a support vertex, then sta(v) = A. 4. If v is a leaf, then sta(v) = B. 5. |A(T)| = |B(T)| 6. Distance between any two vertices in B(T) is at least three. Lemma 3.2 IfT € F, then B(T) is a 7p(T)-set. Moreover if T is obtained from T' € F using operation F, or Fz, then yp(L) = yp(T") +1. Proof By Observation 3.1, it is clear that B(T) is a perfect dominating set. Now we prove that, B(T) is a 7p(T)-set. We proceed by induction on the length n of the sequence of trees needed to construct the tree 7. Suppose n = 1, then T = Fy, belongs to F. Let the vertices of P, be labeled as a,b,c,d. Then, B(P:) = {a, d} and is a yp(P1)-set. This establishes the base case. Assume then that the result holds for all trees in F that can be constructed from a sequence of fewer than n trees where n > 2. Let T € F be obtained from a sequence T,T2,--: , Tp of n trees, where T’ = T,,_-; and T = T,,. By our inductive hypothesis B(T’) is a Yp(T")-set. We now consider two possibilities depending on whether T is obtained from T” by operation Fy, or Fo. Case 1 T is obtained from T’ by operation F,. Suppose T is obtained from J” by adding a path y,z,w of length 2 to the attacher vertex y €V(Z"). Any y,(Z")-set can be extended to a yp(T)-set by adding to it the vertex w, which is of status B. Hence B(T) = B(T’) U {w} is a 7p(T)-set. Case 2 T is obtained from T’ by operation Fp. The proof is very similar to Case 1. If T is obtained from T’ € F using operation F; or Fo, then T can have exactly one more vertex with status B than T’. Since y,(T) = |B(T)| and y,(T") = |B(Z")|, it follows that Yp(T) = yp (T’) +1. Lemma 3.3 IfT € F have length n, then T is ay, — excellent tree. Proof Since T has length n in ¥, T can be obtained from a sequence 7,7 >,--- ,Tp, of trees such that T; is a path Py and T,, = T, a tree. If n > 2, T;,1 can be obtained from T; by one of the two operations F,, Fz fori = 1,2,--- ,n—1. To prove the desired result, we proceed by induction on the length n of the sequence of trees needed to construct the tree T’. Ifn = 1, then T’ = FP, and therefore, T is yp-excellent. Hence the lemma is true for the base case. Perfect Domination Excellent Trees 79 Assume that the result holds for all trees in F of length less than n, where n > 2. Let T € F be obtained from a sequence T),7>,--- ,T, of n trees. For notational convenience, we denote T;,_1 by T’. We now consider two possibilities depending on whether T is obtained from T’ by operation Fy or Fo. Case 1 T is obtained from T’ by operation F,. By Lemma 3.2, y)(T') = yp(T’) + 1. Let wu be an arbitrary element of V(T). Subcase 1.1 ue V(T"). Since T” is yp-excellent, y;(T") = yp(T"). Now any 7; (7")-set can be extended to a perfect dominating set of T by adding either # or w and so f(T) < yp(T") + 1 = y(T") +1 = wp (7). Subcase 1.2 we V(T)\ V(T’). Any 7} (T")-set can be extended to a perfect dominating set of T by adding the vertex w and so 3 (T) < yg(7") +1 = %(T") +1 = (7). Consequently, we have 7;(T’) = y)(T) for any arbitrary vertex u of T. Hence T is 7p- excellent. Case 2 T is obtained from T’ by operation Fo. The proof is very similar to Case 1. Proposition 3.4 [fT is a tree obtained from a tree T’ by adding a path x,w or a path x,w,v and an edge joining x to the vertex y of T’, then y)(T) = yp(Z") +1. Proof Suppose T is a tree obtained from a tree T’ by adding a path z, w and an edge joining x to the vertex y of T’, then any y,(T’)-set can be extended to a perfect dominating set of T by adding x or w and so yp(T) < 7p(T’) +1. Now let S be a y,(T)-set and let S’ = SNV(Z"). Then S” is a perfect dominating set of T’. Hence, y,(T") < |S’| < |S|-—1=7)(T) — 1. Thus, p(T) > p(T") +1. Hence 7)(T) = yp(T") +1. The other case can be proved on the same lines. Theorem 3.5 A tree T of ordern > 4 is yp — excellent if and only if T € F. Proof By Lemma 3.3, it is sufficient to prove that the condition is necessary. We proceed by induction on the order n of a 7Yp-excellent tree T. For n = 4, T = P, is yp-excellent and also it belongs to the family F. Assume that n > 5 and all yp-excellent trees with order less than n belong to F. Let T be a yp-excellent tree of order n. Let P : v1, v2,--+ , vp be a longest path in T. Obviously deg(v1) = deg(v,) = 1 and deg(v2) = deg(vg_1) = 2 and k > 5. We consider two possibilities. Case 1 v3 is a support vertex. Let T’ = T \ {v1, v2}. We prove that T’ is yp-excellent, that is for any u € T’, yp (T") = Yp(T"). Since u € T’ C T and T is yp-excellent, there exists a y/(T)-set such that yp(T) = Yp(T). Let S be a ¥5(T’) — set and S’= SN V(Z"). Then S" is a perfect dominating set of T’. Also, |S"| < |S] — 1 = 7%(T) — 1 = 7(T"), by Proposition 3.4. Since u € S$’, S" is a y}(T")-set 80 Sharada B. such that 7;(Z") = p(T"). Thus T” is yp-excellent. Hence by the inductive hypothesis T’ € F, since |V(T")| < |V(T)|. The sta(v3) = A in T’, because v3 is a support vertex. Thus, T is obtained from T’ € F by the operation F,. Hence T € F as desired. Case 2 v3 is not a support vertex. Let T’ = T \ {v1, v2, v3}. As in Case 1, we can prove that T’ is yp-excellent. Since |V(I")| < |V(T)|, T’ € F by the inductive hypothesis. If v4 is a support vertex or has a neighbor which is a support vertex then v3 is present in none of the yp-sets. So, T’ cannot be 7p-excellent. Hence either deg(vs) = 2 and v4 is a leaf of T’ so that v4 € B(T’) by Observation 3.1 or deg(v4) > 3 and all the neighbors of v4 in T’ \ {vs} are at distance exactly 2 from a leaf of T’. Hence all the neighbors of v4 in T’ \ {vs} are in A(T’) by Observation 3.1, and have no neighbors in B(T’) except v4. Hence v4 € B(T’), again by Observation 3.1. Thus, T can be obtained from T’ by the operation F2. Hence T € F. References 1] H.Aram, S.M.Sheikholeslami and O.Favaron, Domination subdivision numbers of trees, Discrete Math., 309 (2009), 622-628. 2] G.H.Fricke, T.W.Haynes, S.S.Hedetniemi and R.C.Laskar, Excellent trees, Bulletin of ICA, 34(2002) 27-38. 3] T.W.Haynes, S.T.Hedetniemi and P.J.Slater (Eds.), Domination in Graphs: Advanced Top- ics, Marcel Dekker, New York, 1998. 4] T.W.Haynes, S.T.Hedetniemi and P.J.Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998. 5| T.W.Haynes, M.A.Henning, A characterization of i-excellent trees, Discrete Math., 248 (2002), 69-77. 6] M.A.Henning, Total domination excellent trees, Discrete Math., 263(2003), 93-104. 7| D.Sumner, personal communication, May 2000. Math. Combin. Book Ser. Vol.2(2012), 81-88 On (k,d)-Maximum Indexable Graphs and (k,d)-Maximum Arithmetic Graphs Zeynab Khoshbakht Department of Studies in Mathematics University of Mysore, Manasagangothri MYSORE - 570 006, India E-mail: znbkht@gmail.com Abstract: A (n,m) graph G is said to be (k,d) maximum indexable graph, if its vertices can be assigned distinct integers 0,1,2,--- ,2—1, so that the values of the edges, obtained as the sum of the numbers assigned to their end vertices and maximum of them can be arranged in the arithmetic progression k,k + 1,k + 2d,--- ,k+(m-—1)d and also a (n,m) graph G is said to be (k,d) maximum arithmetic graph, if its vertices can be assigned distinct non negative integers so that the values of the edges, obtained as the sum of the numbers assigned to their end vertices and maximum of them can be arranged in the arithmetic progression k,k + 1,k + 2d,---,&+(m-—1)d. The energy E(G) of a graph G is equal to the sum of the absolute values of the eigenvalues of G. In this paper we introduce some families of graphs which are (k, d)- maximum indexable and (k, d)-maximum arithmetic and also compute energies of some of them. Key Words: Graph labeling, indexable graphs, (k, d)- Maximum indexable graphs, (k, d)- maximum arithmetic graphs. AMS(2010): 05078, 05050, 58040 §1. Introduction Let G = (V, FE) be a (n,m) graph and let its vertex set be V(G) = {v1, v2,--+ , Un}. We assume that G is a finite, undirected, connected graph without loops or multiple edges. Graph labelings where the vertices are assigned values subject to certain conditions are interesting problems and have been motivated by practical problems. Applications of graph labeling have been found in X-— ray, crystallography, Coding theory, Radar, Circuit design, Astronomy and communication design. Given a graph G = (V,£), the set N of non-negative integers, a subset A of N of non- negative integers, a set A of N and a Commutative binary operation * : N x N — N every vertex function f : V(G) —> A induces an edge function f* : E(G) — N such that f*(uv) = *(f(u), f(v)) = flu) * fv), Vuu € E(G). We denote such induced map f* of f by fe 1Received February 20, 2012. Accepted June 20, 2012. 82 Zeynab Khoshbakht Acharya and Hedge [1,2] have introduced the concept of indexable and arithmetic graph labelings. Recently present author [7] has introduced the concept of maximum indexable graphs. The adjacency matrix A(G) of the graph G is a square matrix of order n whose (¢, j)-entry is equal to unity if the vertices v; and v; are adjacent, and is equal to zero otherwise. The eigenvalues \1,A2,::: ,An of A(G) are said to be the eigenvalues of the graph G, and are studied within the Spectral Graph Theory [3]. The energy of the graph G is defined as E = E(G) = SOy_, |Ai|.. The graph energy is an invariant much such studied in mathematical and mathema-tico-chemical literature; for details see [4,5,6], In this paper we introduce some families of graphs which are (k,d)- maximum indexable and (k,d)-maximum arithmetic and also compute energies of some of them. Definition 1.1 A graph G = (V,E) is said to be (k,d)— maximum indexable graph if it admits a (k,d)— indexer, namely an injective function f : V(G) —> {0,1,---,n—1} such that f(u) + f(v) + maaf{f(u), f(v)} = fmre"(uv) © f™r(G) = {f™*uv) : Vuv € E(G)} = {k,k+d,k+2d,---,k+(m-—1)d}, for every uw © E. Example 1.2 The graph K2,3 is a (4, 1)—maximum indexable graph. 0 1 Figure 1 In this example, fP""(G) = {4,5, 6, 7, 8, 9}. Lemma 1.3 Let f be any (k,d)—mazimum indexable labeling of G. Then 2<k<3n—md+ (d—4). Proof Since f™°*(G) C {2,3,--- ,38n—4}, the largest edge label is at most 3n — 4. Hence k must be less than or equal to 3n — 4— (m—1)d. Since the edge values are in the set {k,k+d,k+2d,---,k+(m-— 1)d}, we have 2<k<3n—md+ (d—4). On (k, d)-Maximum Indexable Graphs and (k, d)-Maximum Arithmetic Graphs 83 Theorem 1.4 The star Ky, is (k,d)—mazimum indexable if and only if (k,d) = (2,2) or (2n,1). Furthermore, there are exactly n +1 maximum indexable labelings of Ki, of which only two are (k,d)—maximum indexable up to isomorphism . Proof By assigning the value 0 to the central vertex and 1,2,3,---,n to the pendent ver- tices we get (2,2)—maximum indexable graph, since f™**(G) = {2,4,6,--- ,2n}. By assigning the value n to the root vertex and 0,1,2,--- ,n—1 to the pendent vertices, one can see that fm*(G) = {2n, 2n + 1,2n+3,--- ,3n—1}, which shows that G is (2n, 1)-maximum indexable graph. Conversely, if we assign the value c (0 < c < n) to the central vertex and 0,1,2,--- ,c—1,c+ 1,--- ,n to the pendent vertices, we obtain f™* (Ay) = {2c, 2c+1,--+ ,3c—1,3c+2,--- ,2n+c} which is not an arithmetic progressive, i.e., Ky,» is not a (k,d)—maximum indexable. For the second part, note that K,,, is of order n+ 1 and size n. Since there are n+ 1 vertices and n+ 1 numbers (from 0 to n), each vertex of Ay, can be labeled in n + 1 different ways. Observe that the root vertex is adjacent with all the pendent vertices. So if the root vertex is labeled using n + 1 numbers and the pendent vertices by the remaining numbers, definitely the sum of the labels of each pendent vertex with the label of root vertex, will be distinct in all n + 1 Maximum Indexable labelings of Ky,,. It follows from first part that, out of n + 1 maximum indexable labelings of Ky,,, only two are (k,d)—maximum indexable. This completes the proof. Corollary 1.5 The graph G = Ky ,U Kin, n> 1 is (k,d)—maximum indexable graph and its energy is equal to 4\/n — 1. Proof Let the graph G be Ki, U Kin, n > 1. Denote V(G) = {u1, U1; :1l<j< n} U {ua, v2; 21 <j< n}, E(G) = {urv1j : 1 <7 < n}U {ugve; :1 <j <n}. Define a function f : V(G) — {0,1,--- ,2n+1} by f(u1) =2n, f(ug) =2n-1 F(v1j) = (27-1) and f(v2;) = 2G —- 1), for 7 = 1,2,---,n. Thus fP?*(Kan U Kin) = {4n4+ 1,40 4+ 2,4n + 3,-++ ,6n — 2,6n — 1, 6n}. Thus Ky, U Ky,» is (4n +1, 1)-maximum indexable graph. The eigenvalues of Ky, U Ky,» are +J/n—1,+Vn—1,0,0,---,0. Hence E(KinU Kin) =4/n—1. SS 2n—4 times Theorem 1.6 For any integer m > 2, the linear forest F = nP3U mP, is a (2m 4+ 2n,3) maximum indexable graph and its energy is 2(nV/2 +m). 84 Zeynab Khoshbakht Proof Let F = nP3UmP, be a linear forest and V(F) = {ui,vi,wi : 1 <i < n}U {t;,yj : 1 <7 < m}. Then |V(F)| = 3n+ 2m and |E(F)| = 2n+™m. Define f : V(F) — {0,1,---,2m-+3n— 1} by i—1, y= Wi, 1l<i<n, m+tn+(i—1), y=uj, L<i<n, IOyH*. gn 1; y=, 1<j<m, 2n+m)+(i-1), y=w, l<i<n Then PP"(F) = {2(n+m),2(n +m) + 3,2(m+n)+6--- ,2(m+n) + +3(n —1),5n + 2m,--- ,5n+ 5m — 3,5n + 5m, 8n + 5m — 3}. Therefore nP3 UmP3 is a (2(n + m),3)—maximum indexable graph. 0 m+n 2n+2m e—______e+__e 1 m+n+1 2n+2m+1 o_________»—__———_ on m+2n-1 2m+3n-1 o—___——-w————-e ti m+2n ee n+m-1 * 2n+2m-1 o—__—__—_e Figure 2 The eigenvalues of nP3 UU mP, are 1,-1 ,0, V2,—V2. Hence its energy is equal to E(nP3 U KH m times n times mP 2)) = 2(n/2+m). Corollary 1.7 For any integer m > 2, the linear forest F = P3UmPy is a (2m+2,3)—mazimum indexable graph. Proof Putting n = 1 in the above theorem we get the result. On (k, d)-Maximum Indexable Graphs and (k, d)-Maximum Arithmetic Graphs 85 §2. (k,d)— Maximum Arithmetic Graphs We begin with the definition of (k,d)—maximum arithmetic graphs. Definition 2.1 Let N be the set of all non negative integers. For a non negative integer k and positive integer d, a (n,m) graph G = (V,E), a (k,d)—maximum arithmetic labeling is an injective mapping f : V(G) —> N, where the induced edge function f™* : E(G) — {k,k+d,k + 2d,--- ,k+(m—1)d} is also injective. If a graph G admits such a labeling then the graph G is called (k,d)—mazimum arithmetic. Example 2.2 Let $3 be the graph obtained from the star graph with n vertices by adding an edge. $@ is an example of (4,2)—maximum arithmetic graph. 5 Figure 3 S? Here fr (Sey 14.6.8, 10,12, 14). Definition 2.3 Let P, be the path on n vertices and its vertices be ordered successively as £1,22,°** ,2py. P! is the graph obtained from P,, by attaching exactly one pendent edge to each of the vertices 41, %2,°°: , 2X1. Theorem 2.4 P! is (k,d)—mazximum indexable graph. Proof We consider two cases. Case 1 If n is odd. In this case we label the vertices of P! as shown in the Figure . The value of edges can be written as arithmetic sequence {5,8,--- ,(3n—1), (8n + 2), (8n+5),--- ,(8n+ 31 —1)}. It is clear that P! is (5,3)—maximum arithmetic for any odd n. (A) + 0-1) (3et1)41 anda 2 e——————_® n n—1 l I-1 2 1 86 Zeynab Khoshbakht Case 2 If n is even. In this case we label the vertices of P! as shown in the Figure 5. The value of edges can be written as arithmetic sequence {7, 13,--- ,(6n—5), (6n+1), (6n+7), (6n4 13),--- ,(6n + 61 — 5)}. It is clear that P! is (7,6)— maximum arithmetic for any even n. o_ 2n—1 2n—3 1-1 5 3 1 Figure 5 Theorem 2.5 The graph G = KoVKy_2 for n> 3 is a (10,2)— maximum arithmetic graph. Proof Let the vertices of K2 be v; and v2 and those of Ky,_2 be v3, v4,--+ , Un. By letting f(ui) = 2% for i = 1,2 and f(v;) = 22-1, 3 <i <n, we can easily arrange the values of edges of G = KoV K,,_2 in an increasing sequence {10,12,14,--- ,4n,4n + 2}. Lemma 2.6 The complete tripartite graph G = K12,n is a (7,1)—mazimum arithmetic graph. Proof Let the tripartite A, B, C be A = {w}, B = {v1, v2} and C = {v3,v4,--+ , Un}. Define the map f: AU BUC —N by Then one can see that G = Kj1,2, is (7,1)—maximum arithmetic. In fact f™*(Ki2n) = {7,8,9,10,--+,2n+5,2n+6,2n+7,--- ,3n+ 8}. Theorem 2.7 For positive integers m and n, the graph G = mK iy, is a (2mn + m+ 2,1)—maximum arithmetic graph and its energy is 2mV/n— 1. Proof Let the graph G be the disjoint union of m stars. Denote V(G) = {uj:1<i< mbU{uy 2: 1<icmil<y <n} and E(G) = {uwy:1<i<m,l<j <n}. Define f:V(G) — N by f(z) = m(n+1)—-(i-1) ife=v; 1<i<m j}—1)m+i ifw@=uz; 1<i<m,1<j<n. G Jj ys bash Then fP?*(G) = {2mn + m + 2,2mn + m4 3,--- ,38mn+m+ 1h. Therefore mK, is a (2mn + m+ 2,1)—maximum arithmetic graph. On (k, d)-Maximum Indexable Graphs and (k, d)-Maximum Arithmetic Graphs 87 U1 v2 Um Umn UW Um2 U11 U12 Ulin a u22 U2n Um1 Figure 6 Also its eigenvalues are +\/n — 1 (m times). Hence the energy of mK1,,, is E(mKin) = 2mvVn— 1. §3. Algorithm In this section we present an algorithm which gives a method to constructs a maximum (k, d)- maximum indexable graphs. MATRIX-LABELING (Vlist,V size) //Vlist is list of the vertices, V size is number of vertices Elist= Empty(); for j < —0 to (Vsize —1) do for i < —0 to (Vlist —1) && i!=J X=V List[t] + Vlist[j] + Max {Viist[2], Vlist|j]}; if (Search (Elist, X)=False) —_//X does not exist in Elist then ADD (Elist, X); end if end for end for SORT-INCREASINGLY (Elist); Flag= 0; For j < —0 to (Esize — 1) do if (Elist[j + 1]! = Elist{j] + d) then Flag= 1; end if end for if (Flag== 0) 88 Zeynab Khoshbakht then PRINT (“This graph is a (k, d)—Max indexable graph” ); else if (Flag= 1) then PRINT (“This graph is not (&,d)—Max indexable graph” ); end if end MATRIX-LABELING(Vlist, V size) References 1 B.D.Acharya and S.M.Hegde, Strongly indexable graphs, Discrete Math., 93 (1991) 123- 129. B.D.Acharya and S.M.Hegde, Arithmetic graphs, J.Graph Theory, 14(3) (1990), 275-299. D.Cvetkovié, M.Doob and H.Sachs, Spectra of Graphs- Theory and Application, Academic Press, New York, 1980. I.Gutman, The energy of a graph: Old and new results, in: A. Betten, A. Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer Verlag, Berlin, (2001), 196-211. I.Gutman, Topology and stability of conjugated hydrocarbons. The dependence of total m—electron energy on molecular topology, J. Serb. Chem. Soc., 70 (2005), 441-456. I.Gutman and O.E.Polansky, Mathematical Concepts in organic Chemistry, Springer Ver- lag, Berlin, 1986. Z.Khoshbakht, On Maximum indexable graphs, Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 31, 1533-1540. Math.Combin. Book Ser. Vol.2(2012), 89-95 Total Dominator Colorings in Paths A. Vijayalekshmi (S.T.Hindu College, Nagercoil, Tamil Nadu, India) E-mail: vijimath.a@gmail.com Abstract: Let G be a graph without isolated vertices. A total dominator coloring of a graph G is a proper coloring of the graph G with the extra property that every vertex in the graph G properly dominates a color class. The smallest number of colors for which there exists a total dominator coloring of G is called the total dominator chromatic number of G and is denoted by xza(G). In this paper we determine the total dominator chromatic number in paths. Unless otherwise specified, n denotes an integer greater than or equal to 2. Key Words: Total domination number, chromatic number and total dominator chromatic number, Smarandachely k-domination coloring, Smarandachely k-dominator chromatic num- ber. AMS(2010): 05C15, 05C69 §1. Introduction All graphs considered in this paper are finite, undirected graphs and we follow standard defini- tions of graph theory as found in [2]. Let G = (V,E£) be a graph of order n with minimum degree at least one. The open neighborhood N(v) of a vertex v € V(G) consists of the set of all vertices adjacent to v. The closed neighborhood of v is N[v] = N(v) U {v}. For a set S C V, the open neighborhood N(S) is defined to be UyegN(v), and the closed neighborhood of S$ is N[S] = N(S)US. A subset S' of V is called a dominating (total dominating) set if every vertex in V — S (V) is adjacent to some vertex in S. A dominating (total dominating) set is minimal dominating (total dominating) set if no proper subset of S is a dominating (total dominating) set of G. The domination number y (total domination number 7) is the minimum cardinality taken over all minimal dominating (total dominating) sets of G. A +y-set (7-set) is any minimal dominating (total dominating) set with cardinality 7 (44). A proper coloring of G is an assignment of colors to the vertices of G, such that adjacent vertices have different colors. The smallest number of colors for which there exists a proper col- oring of G is called chromatic number of G and is denoted by x(G). Let V = {u1, wa, us,--- , Up} and C = {C, C2, C3,--- ,C,} be a collection of subsets C; C V. A color represented in a vertex u is called a non-repeated color if there exists one color class C; € C such that C; = {u}. Let G be a graph without isolated vertices. A total dominator coloring of a graph G is 1Received September 19, 2011. Accepted June 22, 2012. 90 A. Vijayalekshmi a proper coloring of the graph G with the extra property that every vertex in the graph G properly dominates a color class. The smallest number of colors for which there exists a total dominator coloring of G is called the total dominator chromatic number of G and is denoted by xXta(G). Generally, for an integer k > 1, a Smarandachely k-dominator coloring of G is a proper coloring on G such that every vertex in the graph G properly dominates a k color classes and the smallest number of colors for which there exists a Smarandachely k-dominator coloring of G is called the Smarandachely k-dominator chromatic number of G, denoted by xX2(G). Clearly, if k; = 1, such a Smarandachely 1-dominator coloring and Smarandachely 1-dominator chromatic number are nothing but the total dominator coloring and total dominator chromatic number of G. In this paper we determine total dominator chromatic number in paths. Throughout this paper, we use the following notations. Notation 1.1 Usually, the vertices of P,, are denoted by uj, u2,...,Un in order. We also denote a vertex uj © V(P,) with 1 > [$] by w_(n41). For example, un—1 by u_z. This helps us to visualize the position of the vertex more clearly. Notation 1.2 Fori < j, we use the notation ((7, j]) for the subpath induced by (uj, wit, ..., Uy)- For a given coloring C' of P,,, C|([t, j]) refers to the coloring C restricted to ([i, j]). We have the following theorem from [1]. Theorem 1.3 For any graph G with 6(G) > 1, max{y(G),7%4(G)} < xta(G) < x(G) + 74(G). Definition 1.4 We know from Theorem 1.3 that yta(Pn) © {ye(Pn), Ve(Pr) + 1, %4(Prn) + 2}- We call the integer n, good (respectively bad, very bad) if Xta(Pn) = Ye(Pn) +2 (if respectively Xta(Pr) = Ye(Pn) +1, xta(Pr) = Ye(Pr))- §2. Determination of yta(P,) First, we note the values of yta(P,,) for small n. Some of these values are computed in Theorems 2.7, 2.8 and the remaining can be computed similarly. Total Dominator Colorings in Paths 91 Thus n = 2,3,6 are very bad integers and we shall show that these are the only bad integers. First, we prove a result which shows that for large values of n, the behavior of x+a(P,) depends only on the residue class of nmod 4 [More precisely, if n is good, m > n and m = n(mod 4) then m is also good]. We then show that n = 8,13,15,22 are the least good integers in their respective residue classes. This therefore classifies the good integers. Fact 2.1 Let 1 <i<_nand let C be a td-coloring of P,. Then, if either u; has a repeated color or u;+2 has a non-repeated color, C|([¢ + 1,n]) is also a td-coloring. This fact is used extensively in this paper. Lemma 2.2) yta(Pni4) > xXta(Pn) + 2. Proof For 2 < n < 5, this is directly verified from the table. We may assume n > 6. Let uj, U2,U3,.--,Un+4 be the vertices of P,44 in order. Let C’ be a minimal td-coloring of Py+4. Clearly, ug and w_2 are non-repeated colors. First suppose u4 is a repeated color. Then C| ([5,n + 4]) is a td-coloring of P,. Further, C| ([1, 4]) contains at least two color classes of C. Thus xta(Pn + 4) > xvta(Pr) + 2. Similarly the result follows if u_4 is a repeated color Thus we may assume u4 and w_4 are non-repeated colors. But the C| ([3, + 2]) is a td-coloring and since Ug and u_»2 are non-repeated colors, we have in this case also xta(Pni4) > xta(Pn) + 2. Corollary 2.3 If for any n, xta(Pn) = ¥2(Pn) + 2, Xta(Pm) = ¥%4(Pmn) +2, for all m > n with m = n(mod 4). Proof By Lemma 2.2, xta(Pn+4) > Xta(Pn) + 2 = %(Pr) + 2+ 2 = %4(Pr4+a) + 2. Corollary 2.4 No integer n > 7 is a very bad integer. Proof For n = 7,8,9,10, this is verified from the table. The result then follows from the Lemma 2.2. Corollary 2.5 The integers 2,3,6 are the only very bad integers. Next, we show that n = 8,13,15,22 are good integers. In fact, we determine yta(P,,) for small integers and also all possible minimum td-colorings for such paths. These ideas are used more strongly in determination of yza(P,) for n = 8,13, 15, 22. Definition 2.6 Two td-colorings C, and C2 of a given graph G are said to be equivalent if there exists an automorphism f : G— G such that Co(v) = Ci(f(v)) for all vertices v of G. This is clearly an equivalence relation on the set of td-colorings of G. 92 A. Vijayalekshmi Theorem 2.7 Let V(P,) = {u1, u2,...,Un} as usual. Then (1) Xta(P2) = 2. The only minimum td-coloring is (given by the color classes) {{u1}, {u2}} (2) xXta(P3) = 2. The only minimum td-coloring is {{ui,u3}, {ua}}. (3) xXta(P1) = 3 with unique minimum coloring {{u1, ua}, {uz}, {us}}. (4) xXta(P5) = 4. Any minimum coloring is equivalent to one of {{ui, us}, {ua}, {ua}, {us}} or {{u1, us}, {uz}, {us}; {ua}} or {{ui}, {ug}, {ua}, {us, us}}- (5) xXta(Ps) = 4 with unique minimum coloring {{u1, us}, {ua, ue}, {uz}, {us}}. (6) xta(P7) = 5. Any minimum coloring is equivalent to one of {{u1, us}, {uz}, {ua, uz}, {us}, {ue}} or {{u1, us}, {uz}, {us}, {us, ur}, {ue}} {{u1, Ua, ur}, {uz}, {us}, {us}, {ue}}- 7 Proof We prove only (vi). The rest are easy to prove. Now, %(P7) = [=] = 4. Clearly Xta(P7) > 4. We first show that x:q(P7) 4 4 Let C be a td-coloring of P; with 4 colors. The ver- tices ug and u_z2 = ug must have non-repeated colors. Suppose now that us has a repeated color. Then {ui, u2, ug} must contain two color classes and C| ([4,7]) must be a td-coloring which will require at least 3 new colors (by (3)). Hence w3 and similarly u_3 must be non-repeated colors. But, then we require more than 4 colors. Thus y+a(P7) = 5. Let C be a minimal td-coloring of P;. Let wz and u_2z have colors 1 and 2 respectively. Suppose that both uz and u_3 are non-repeated colors. Then, we have the coloring {{u1, w4, uz}, {uo}, {us}, {us}, {ue}}. If either ug or u_3 is a repeated color, then the coloring C’ can be verified to be equivalent to the coloring given by {{u1, us}, {ug}, {ua, uz}, {us}, {ue}}, or by {{u1, ua}, {uz}, {us}, {us, ur}, {ue}}- We next show that n = 8,13, 15,22 are good integers. Theorem 2.8 x¢a(Pn) = %(Pr) +2 ifn = 8, 13,15, 22. Proof As usual, we always adopt the convention V(P,,) = {u1,U2,...,Un};U-i = Un4i—i for i > lel C denotes a minimum td-coloring of Py. We have only to prove |C| > (Pn) +1. We consider the following four cases. Case l n=8 Let |C| = 5. Then, as before uz, being the only vertex dominated by uz has a non-repeated color. The same argument is true for u_2 also. If now ug has a repeated color, {u1, u2, u3} contains 2-color classes. As C|([4, 8]) is a td-coloring, we require at least 4 more colors. Hence, ug and similarly u_3 must have non-repeated colors. Thus, there are 4 singleton color classes and {ug}, {u3}, {u-2} and {u_3}. The two adjacent vertices u4 and u_4 contribute two more colors. Thus |C| has to be 6. Case 2 n=13 Let |C| = 8 = %(Pi3)+1. As before ug and u_2 are non-repeated colors. Since yta(Pio) = 7+2 = 9, uz can not be a repeated color, arguing as in case (i). Thus, ug and u_3 are also non- repeated colors. Now, if u; and u_; have different colors, a diagonal of the color classes chosen Total Dominator Colorings in Paths 93 as {u1, U-1, U2, U_2, U3, U_3,---} form a totally dominating set of cardinality 8 = y(Pi3) + 1. However, clearly u; and u_; can be omitted from this set without affecting total dominating set giving %(Pi3) < 6, a contradiction. Thus, u; and u_; = ui3 have the same color say 1. Thus, ({4,—4]) = ([4,10]) is colored with 4 colors including the repeated color 1. Now, each of the pair of vertices {w4, ue}, {us, u7}, {us, Wig} contains a color classes. Thus ug = u_5 must be colored with 1. Similarly, us. Now, if {u4, ug} is not a color class, the vertex with repeated color must be colored with 1 which is not possible, since an adjacent vertex us which also has color 1. Therefore {u4, ug} is a color class. Similarly {ug,wio} is also a color class. But then, uz will not dominate any color class. Thus |C| = 9. Case 3 n=15 Let |C| = 9. Arguing as before, wz, u_2, ug and u_3 have non-repeated colors [V1a(Pi2) = 8]; ui; and u_y have the same color, say 1. The section ([4,—4]) = ([4,12]) consisting of 9 vertices is colored by 5 colors including the color 1. An argument similar to the one used in Case (2), gives u4 (and w_4) must have color 1. Thus, C| ([5, —5]) is a td-coloring with 4 colors including 1. Now, the possible minimum td-coloring of P7 are given by Theorem 2.7. We can check that 1 can not occur in any color class in any of the minimum colorings given. e.g. take the coloring given by {us, us}, {ue}, {uz}, {uo, wir}, {uo}. If ug has color 1, us can not dominate a color class. Since u4 has color 1, {us, ug} can not be color class 1 and so on. Thus Xtd(Pis) = 10. Case 4 n= 22 Let |C| = %(Po2) + 1 = 13. We note that x1a(Pi9) = %4(Pi9) + 2 = 12. Then, arguing as in previous cases, we get the following facts. Fact 1 u2,u_—2, U3, u-3 have non-repeated colors. Fact 2 u; and u_, have the same color, say 1. Fact 3 wu7 is a non-repeated color. This follows from the facts, otherwise C| ([8,22]) will be a td-coloring; The section ([1, 7]) contain 4 color classes which together imply x+a(Po2) > 4+xX+ta(Pis) = 4+10 = 14. In particular {us, uz} is not a color class. Fact 4 The Facts 1 and 2, it follows that C| ({4,—4]) = C| ([4,19]) is colored with 9 colors including 1. Since each of the pair {{ua, uo}, {us, U7}, {us, io}, {uo, Wii, {ur2, Uist, {u13, Wis }, {u16, vis}, {u17, uig}} contain a color class, if any of these pairs is not a color class, one of the vertices must have a non-repeated color and the other colored with 1. From Fact 3, it then follows that the vertex us must be colored with 1. It follows that {u4, ug} must be a color class, since otherwise either u4 or ug must be colored with 1. Since {ua, ue} is a color class, u7 must dominate the color class {ug}. We summarize: © U2, U3, U7, Ug have non-repeated colors. e {u4, ue} is a color class 94 A. Vijayalekshmi e wu; and us are colored with color 1. Similarly, © U_2,U_3, U_7, U_g have non-repeated colors. e {u_4, u_e} is a color class. e u_; and u_s are colored with color 1. Thus the section ([9, —9]) = ({9,14]) must be colored with 3 colors including 1. This is easily seen to be not possible, since for instance this will imply both ui3 and ui4 must be colored with colorl. Thus, we arrive at a contradiction. Thus yza(P22) = 14. Theorem 2.9 Let n be an integer. Then, (1) any integer of the form 4k, k > 2 is good; (2) any integer of the form 4k +1, k > 3 is good; (3) any integer of the form 4k + 2, k (4) any integer of the form 4k +3, k > 5 is good; > 3 is good. Proof The integers n = 2,3,6 are very bad and n = 4,5,7,9,10,11, 14, 18 are bad. Remark 2.10 Let C be a minimal td-coloring of G. We call a color class in C, a non-dominated color class (n-d color class) if it is not dominated by any vertex of G. These color classes are useful because we can add vertices to these color classes without affecting td-coloring. Lemma 2.11 Suppose n is a good number and P, has a minimal td-coloring in which there are two non-dominated color class. Then the same is true forn+A4 also. Proof Let C1, C2,...,C; be the color classes for P,, where C, and C2 are non-dominated color classes. Suppose u,, does not have color Cy. Then CyU{un+1}, CoU{unsa}, {Une}, {un+3}, C3, C4,--- ,C, are required color classes for P,+44. i.e. we add a section of 4 vertices with mid- dle vertices having non-repeated colors and end vertices having C, and C2 with the coloring being proper. Further, suppose the minimum coloring for P,,, the end vertices have different colors. Then the same is true for the coloring of P,,44 also. If the vertex u, of P,, does not have the color C2, the new coloring for P,44 has this property. If wu; has color C2, then u,, does not have the color C2. Therefore, we can take the first two color classes of Pr44 as Cy U {un+a} and C2 U {un41}. Corollary 2.12 Letn be a good number. Then P, has a minimal td-coloring in which the end vertices have different colors. [It can be verified that the conclusion of the corollary is true for alln 4 3,4,11 and 18]. Proof We claim that P, has a minimum td-coloring in which: (1)there are two non- dominated color classes; (2)the end vertices have different colors. n=8 n=13 n=15 n = 22 Now, it follows from the Lemma 2.11 that (1) and (2) are true for every good integer. O Total Dominator Colorings in Paths 95 e 2 -2- -2- 2- 2- 2 2 nl 2 2 2 ad Ty T2 TL T2 TL r2 e 2s 2s 2s a ad ad ad nd nd ad ad nd Ty T2771 T2 TI. T2 Fig.1 Corollary 2.13 Let n be a good integer. Then, there exists a minimum td-coloring for P, with two n-d color classes. References 1] M.I. Jinnah and A.Vijayalekshmi, Total Dominator Colorings in Graphs, Ph.D Thesis, Uni- versity of Kerala, 2010. 2| F.Harrary, Graph Theory, Addition - Wesley Reading Mass, 1969. 3] Terasa W.Haynes, Stephen T.Hedetniemi, Peter J.Slater, Domination in Graphs, Marcel Dekker , New York, 1998. 4] Terasa W.Haynes, Stephen T.Hedetniemi, Peter J.Slater, Domination in Graphs - Ad- vanced Topics, Marcel Dekker,New York, 1998. Math.Combin. Book Ser. Vol.2(2012), 96-102 Degree Splitting Graph on Graceful, Felicitous and Elegant Labeling P.Selvaraju 1, P.Balaganesan?, J.Renuka ? and V.Balaj 4 1. Department of Mathematics, Vel Tech Engineering College,Chennai- 600 062, Tamil Nadu, India 2. Department of Mathematics, Saveetha School of Engineering, Saveetha University, Chennai- 602 105, Tamil Nadu, India 3. Departments of Mathematics, Sai Ram College of Engineering, Chennai - 600 044, Tamil Nadu, India 4. Department of Mathematics, Sacred Heart College(Autonomous) Tirupattur-635 601, Vellore(Dt), Tamil Nadu, India E-mail: balki2507@yahoo.co.in Abstract: We show that the degree splitting graphs of Bnjn; Pr; Kmjnj n(ka — 3e)I; n(ka — 3e)I1(b); n(ka — e)IT and n(ka — 2e)II(a) are graceful [3]. We prove C30K1,n is graceful, felicitous and elegant [2], Also we prove K2,n is felicitous and elegant. Key Words: Degree splitting graph, graceful graph, elegant graph, felicitous graph, star and path. AMS(2010): 05C78 §1. Introduction Graph labeling methods were introduced by Rosa in 1967 or one given by Graham and Sloane in 1980. For a graph G, the splitting graph S(G) is obtained from G by adding for each vertex v of G, a new vertex v’ so that vu’ is adjacent to every vertex in G. Let G be a graph with qg edges. A graceful labeling of G is an injection from the set of its vertices into the set {0,1,2,---q} such that the values of the edges are all the numbers from 1 to q, the value of an edge being the absolute value of the difference between the numbers attributed to their end vertices. In 1981 Chang, Hiu and Rogers defined an elegant labeling of a graph G, with p vertices and q edges as an injective function from the vertices of G to the set {0,1,2,---q} such that when each edge xy is assigned by the label f(x) + f(y)(mod (q+ 1)), the resulting edge labels are distinct and non zero. Note that the elegant labeling is in contrast to the definition of a harmonious labeling [1]. Another generalization of harmonious labeling is felicitous labeling. An injective function f from the vertices of a graph G with q edges to the set {0,1,2,---q} is called felicitous labeling if the edge label induced by f(x) + f(y)(mod q) for each edge zy is distinct. 1Received January 30, 2011. Accepted June 24, 2012. Degree Splitting Graph on Graceful, Felicitous and Elegant Labeling 97 §2. Degree Splitting Graph DS(G) Definition 2.1 Let G = (V,E) be a graph with V = S$, U S2:U $3 U---S;UT where each S; is a set of vertices having at least two vertices of the same degree and T =V\US;. The degree splitting graph of G denoted by DS(G) is obtained from G by adding vertices w1, w2,W3,-+++ , We and joining to each vertex of S; for 1 <i<t. §3. Main Theorems Theorem 3.1 The DS(Bn.») ts graceful for n > 2. Proof Let G = By» be a graph.Let V(G) = {u,v,ui,u; : 1 <i <n} and V(DS(G)) \ V(G) = {wi, we}. Let E(DS(G)) = {uv, wwe, vwe} U {uui, vu;, wiui, Wi; 2 1 <i <n} and | E(DS(G)) |= 4n 4+ 3. The required vertex labeling f : V(DS(G)) > {0,1,2,--- ,4n + 3} is as follows: f(u) =1; f(v) = 38; f(w1) = 0; f(we) = 2n + 4; f(us) = 4n — 21 +5 and f(v;) = 274 2 for l<i<n. The corresponding edge labels are as follows: The edge label of wv is 2; wu; is 4n — 204+ 4 for 1 <i <n; vv; is 2i-—1 for l <i<njs wy; is 4n — 214+ 5 for 1 <i < nj wiv; is 214+ 2 for 1 <i < nj uwe is 2n+ 3 and vwe is 2n + 1. Hence the induced edge labels of DS(G) are 4n + 3 distinct integers. Hence DS(G) is graceful for n > 2. Theorem 3.2 The DS(P,) is graceful for n > 4. Proof Let G = P, be a graph. Let V(G) = {v; : 1 <i < n} and V(DS(G)) \ V(G) = {wi, wa}. Let E(DS(G)) = {wivi, win} U {wovj : 2<4<n-—1} and | E(DS(G)) |= 2n-1. The required vertex labeling f : V(DS(G) — {0,1,2,--- ,2n— 1} is as follows: Case 1 nis odd. Then f(wi) =n+1; f(we) = 0; f(u) =i for 1<i<n, iis odd and f(v;) = 2n-i+1 for 1<i<n, 7 is even. The corresponding edge labels are as follows: The edge label of wou; is i for 3 <1 <n-—1 and i is odd; wav; is 2n-i+1 forl<i<n and 7 is even; w V1 is nN; W1Vy, is 1 and v;V;41 is 2n — 2i for 1 <7 <n-—J1. Hence the induced edge labels of DS(G) are 2n — 1 distinct integers. Hence DS(G) for n > 4 is graceful. Case 2 n is even. The required vertex labeling is as follows: f(wi) = n+ 2; f(we) = 0; f(u;) = 7 for 1 <i < n,i is odd and f(u;) = 2n-i+1 for 98 P.Selvaraju, P.Balaganesan, J.Renuka and V.Balaj 1<i< 7,2 is even. The corresponding edge labels are as follows: The edge label of w2v,; is 7 for 3 <7 <n and 7 is odd; wou; is 2n-i+1 forl<i<n-1 and 7 is even; w,v1 is N+1; win is 1 and v,vj41 is 2n—2 for 1 <<71<n-—1. Hence the induced edge labels of G are 2n — 1 distinct integers. From case (i) and (ii) the DS(P,,) for n > 4 is graceful. Theorem 3.3 The graph DS(Km.n) is graceful. Proof The proof is divided into two cases following. Case l m>n. Let G = Ky» be a graph. Let V(G) = {u;: 1 <i < mb}Uf{uj:1 <7 < n} and V(DS(G)) \ V(G) = {wi, we}. Let E(DS(G)) = {wus 1 <i < mb}Uf{wo:1<j< n}U {uivj :1<i<m,1<j <n} and | E(DS(G)) |=mn+m-+n. The required vertex labeling f : V(DS(G)) > {0,1,2,---,mn+m-+n} is as follows: f(u) = m(n+2-i)4+n for 1 <i < mf(v;) = 7 for 1 < 7 < n;f(wi) = 0 and f(w2) =n+1. The corresponding edge labels are as follows: The edge label of wi u; is m(n+2—1)+n for 1 <i< m;ujv; is mM(n+2—1)+n—j for 1<icm1l<j<nand wov; isn+1—j for 1 <j <n. Hence the induced edge labels of G aremn+m-+n distinct integers. Hence the graph DS(K mn) is graceful. Case 2 m=n. Let G = Kmn be a graph Let V(G) = {wu : 1 < it < m} and V(DS(G))\ V(G) = {ui}. Let E(DS(G)) = {wiwi, wivi: 1<i< m}U {ujv; +1 < 14,7 < m} and | E(DS(G)) |= m(m + 2). The required vertex labeling f : V(DS(G)) — {0,1,2,--- ,m(m + 2)} is as follows: f(wi) = 0; f(us) = m(m + 3) — mi for 1 <i < mand f(v;) =i for 1<i<m. The corresponding edge labels are as follows: The edge label of wiv; is i for 1 < i < m;ujv; is mM(m+ 3) — mi — 7 for 1 < i,j < mand wu; is m(m+3)—mi for 1 <i <m. Hence the induced edge labels of G are m(m-+ 2) distinct integers. Hence the graph DS(Kym,n) is graceful. Corollary 3.4 The DS(K,,) is Ky41. Theorem 3.5 The DS(n(K4 — 3e)1)) is graceful. Proof Let G = n(K4 — 3e)I be a graph. Let V(G) = {a,y}U{z:1 <i < n} and V(DS(G)) \ V(G) = {wi, wo}. Let E(DS(G)) = {xwe, ywo, ry} U {wizi, v2, yz 1<i<n} Degree Splitting Graph on Graceful, Felicitous and Elegant Labeling 99 and | E(DS(G)) |= 3n+3. The required vertex labeling f : V(DS(G)) > {0,1,2,--- ,3n+ 3} is as follows: f(x) =1; fly) = 2; f(w1) = 0; f(we) = 4 and f(z;) = 3n + 6-37 for l<i<n. The corresponding edge labels are as follows: The edge label of ry is 1; rw is 3; ywe is 2; xz; is 8n+5— 31 for 1 <1 < nj yz; is 8n+4-—31 for 1 <i <nand w,zZ; is 3n + 6 — 32 for 1 < i < n. Hence the induced edge labels of G are 3n +3 distinct integers. Hence the DS(G) is graceful. Theorem 3.6 The DS((n(K4 — 3e)II(b)) is graceful. Proof Let G = n(K4 — 3e)II(b) be a graph. Let V(G) = {x,y} U {ui,u,: 1 <i <n} and V(DS(G)) \ V(G) = {w1, wo} Let E(DS(G)) = {aw1, cy} U {wij;, wii, yur, W2uj 2 1 <i <n} and | E(DS(G)) |= 4n+ 2. The required vertex labeling f : V(DS(G)) > {0,1,2,--- ,4n + 2}is as follows: f(x) = 38n + 2; f(y) = 1; f(wi) = 0; f(we) = 2; fui) = 38n+2+7% for 1 <i < n and f(u;) = 20+ 1 forl<i<n. The corresponding edge labels are as follows: The edge label of w vu; is 8n+2+i7 for 1<i<njujv; is 8n—-—i4+1 for 1 <i< n;yu; is 22 for 1 <i <n;weu; is 2i-—1 for 1 <i < nj rw, is 8n+ 2 and zy is 3n+ 1. Hence the induced edge labels of G are 4n + 2 distinct integers. Hence the graph DS(G) is graceful. Theorem 3.7 The DS(n(K4— e)II) is graceful. Proof Let G = n(K4 — e)ITI) be a graph. Let V(G) = {z,y} U {ui,u; : 1 <i < nm} and (DS(G)) \ V(G) = {w1, wo} Let E(DS(G)) = {xwe, ywo} U {wiui, wiv; for 1 <7 < n} and | E(DS(G)) |= 6n + 3. The required vertex labeling f : V(DS(G)) > {0,1,2,--- ,6n + 3} is as follows: f(x) = 0; f(y) = 4n + 2; f(wi) = 2n + 2; f(we) = 4n + 3; f(ui) = 5n+4—i and f(ui) = 6n+4—-iforl<i<n. The corresponding edge labels are as follows: The edge label of w,u; is 4n + 2-7 for 1 <i < n;w vy; is dn- 1-1 for 1 <2 < nj 2we is 4n+ 3; zy is 4n + 2 and ywe is 2n. Hence the induced edge labels of G are 6n + 3 distinct integers. The DS(n(K4 — e)IT) is graceful. Theorem 3.8 The DS(n(K4— 2e)II(a)) is graceful. Proof Let G = n(K4 — 2e)II(a) be a graph. Let V(G) = {z,y,ui,u,: 1 < i < n} and V(DS(G)) \ V(G) = {wi, wo}. Let E(DS(G)) = {ui cy, yui, cu; viW1, Uwe 21 <i<n} and | E(DS(G)) |= 5n+1. 100 P.Selvaraju, P.Balaganesan, J.Renuka and V.Balaj The required vertex labeling f : V(DS(G)) > {0,1,2,--- ,5+ 1} is as follows: f(x) = 0; f(y) = 2n+1; f(wi) = 1; f(w2) = n4+1; f(vi) = 5n4+3-2i and f(u;) = 38n+2-1 forl<i<n. The corresponding edge labels are as follows: The edge label of xu; is 3n—71+2 for 1 <i < nj ry is 2n+1; yu; isn—i+1 for l1<i< nj; xv; is 5n+3-— 22 for 1 <i <n; u;w, is 5n4+ 2-27 for 1 <i <nand ujwe is 2n—714+1 for 1 <i<n. Hence the induced edge labels of G are 5n + 1 distinct integers. The DS(n(K4 — 2e)II(a))) is graceful. Theorem 3.9 The DS(C30K, n) is graceful for n > 3. Proof Let graph G = C30Kin be a graph. Let V(K1,,) = {z} U{uj:1<7i<n} and Cs be the cycle xyzx. Let V(DS(G))\V(G) = {w1, we}. Let E(DS(G)) = {awe, ywo, ry, yz, zx}U {witi, zu; :1<i<njsand | E(DS(G)) | = 2n+5. The required vertex labeling f : V(DS(G)) > {0,1,2,--- ,2+ 5} is as follows: f(w:) = 15 f(we) = 2 fle) = 4; fly) =5; f(z) = 0 and f(ui) =28+5 for 1<i <n. The corresponding edge labels are as follows: The edge label of xy is 1; xwe is 2; yw is 3; x22 is 4; yz is 5; zu; is 21+ 5 for 1 <i<nand wu; is 2i+4 for 1 <2i<n. Hence the induced edge labels of G are 2n + 5 distinct integers. Hence the DS(G) is graceful for n > 3. Theorem 3.10 The DS(C30K,.n) is felicitous when n > 3. Proof Let G = C30 Kin be a graph. Let V(Ki») = {z} U{u;: 1 <i <n} and C3 be the cycle xyzx. Let V(DS(G)) \ V(G) = {wi, wo}. Let E(DS(G)) = {rwe, ywo, ry, yz, zz} U {witi, zu; :1<i <n} and | E(DS(G)) |= 2n+5. The required vertex labeling f : V(DS(G)) > {0,1,2,--- ,2+.5} is as follows: f(wi) =n; f(w2) = 2n + 2; f(x) = 2n4+ 3; f(y) = 2n4+ 4; f(z) = 2n+5 and f(u;) =i-1 forl<i<n. The corresponding edge labels are as follows: The labels of the edges xy is (4n + 7)(mod 2n + 5); xw2 is (4n+5)(mod 2n+ 5); ywe is (4n + 6)(mod 2n +5); az is (4n+ 8)(mod 2n+ 5); yz is (4n+9)(mod 2n +5); zu; is 7 — 1 for 1<i<nand wy u; isn+i-—1 for 1<7i<n. Hence the induced edge labels of G are 2n + 5 distinct integers. Hence the DS(C3OK,,») is a felicitous for n > 3. Theorem 3.11 The DS(C30K in is elegant for n> 3. Proof Let G = C30 Kin be a graph. Let V(Ki.,) = {z}U{u;: 1 <i <n} and C3 be the cycle xyzx and V(DS(G)) \ V(G) = {wi, we}. Let E(DS(G)) = {xwe, ywe, ry, yz, zz} U Degree Splitting Graph on Graceful, Felicitous and Elegant Labeling 101 {witi, zu; :1<i <n} and | E(DS(G) |= 2n+5. The required vertex labeling f : V(DS(G)) > {0,1,2,--- , 2+ 5} is as follows: f(w1) = n+ 2; f(we) = 2n4+ 4; f(x) =n+3; f(y) = 0; f(z) = 2n+5 and f(u;) =i +1; for 1l<i<n. The corresponding edge labels are as follows: The edge labels of ry is (n+ 3); ewe is (8n + 7)(mod 2n + 6); ywe is (2n + 4)(mod 2n + 6); xz is (3n + 8)(mod 2n + 6); yz is (2n + 5)(mod 2n + 6); zu; is (2n + 6 + 4)(mod 2n + 6) for 1L<i<n; wiu; is (n+3+%)(mod2n+6) for1<i<n. Hence the induced edge labels of G are 2n + 5 distinct integers. Hence the DS(G) is elegant for n > 3. Theorem 3.12 The DS(K2,,) is felicitous for n> 3. Proof Let G = (K2,,) be a graph. Let V(G) = {v1, va} U{uj: 1 <i <n} and V(DS(G))\ V(G) = {wi, we}. Let E(DS(G)) = {wov1, wove} U {wiu; : 1 <i < n} and | E(DS(G)) |= 3n + 2. The required vertex labeling f : V(DS(G)) > {0,1,2,--+ ,3m+ 2} is as follows: f(ui) = 3n+ 5 — 3% for 1 <i <n; f(vi) = 0; f(v2) = 8n + 1; f(wi) = 1 and f(we) = 3. The corresponding edge labels are as follows: The edge label of wyu; is (8n + 6 — 32)(mod 3n 4+ 2) forl < i < niujvi is (Bn +5 —-— 3i)(mod 3n + 2) for 1 <i < n; uve is (6n + 6 — 3t)(mod 3n + 2) for 1 <i < nj; wevy is 3 and wev2 is 3n + 4(mod 3n + 2). Hence the induced edge labels of G are 3n + 2 distinct integers. Hence DS(K2,,) is felicitous for n > 3. Theorem 3.13 The DS(K2,,) is elegant for n > 3. Proof Let G = (K2,,) be a graph, LetV(G) = {v1, vo} U {uj : 1 <i < nm} and V(DS(G)) \ V(G) = {wi, we}. Let E(DS(G)) = {wov1, wove} U {wiu; : 1 <i < n} and | E(DS(G)) |= 3n + 2. The required vertex labeling f : V(DS(G)) > {0,1,2,--- ,3n+ 2} is as follows: f(ui) = 38n+5— 3% for 1 <i<n;f(v1) = 0; f(ve) = 3n +1; f(w1) = 2 and f(we) = 4. The corresponding edge labels are as follows: The edge label of w iu; is (8n + 7 — 3i)(mod 3n + 3) for 1 < i < njujy is (8n +5 — 31)(mod 3n + 3) for 1 <i <n; ujve is (6n + 6 — 32)(mod 3n 4+ 3) for 1 <i <n; wevy is 4 and wev2 is 3n +5 (mod 3n +3) . Hence the induced edge labels of G are 3n + 2 distinct integers .The DS(G) is elegant for n > 3. 102 P.Selvaraju, P.Balaganesan, J.Renuka and V.Balaj References [1] J.A.Gallian, A dynamic survey of graph labeling, The Electronic. J. Combinatorics, 16(2009), ##DS6. [2] E.Sampathkumar and Walikar, On the splitting graph of a graph, The Karnataka Univer- sity Journal Science, 25(1980), 13-16. [3] Sethuraman G. and Selvaraju.P, New classes of graphs on graph labeling, National Con- ference on Discrete Mathematics and the Applications, Thirunelveli, January 5-7, 2000. Math. Combin. Book Ser. Vol.2(2012), 103-109 Distance Two Labeling of Generalized Cacti S.K.Vaidya (Saurashtra University, Rajkot - 360005, Gujarat, India) D.D.Bantva (L.E. College, Morvi - 363642, Gujarat, India) E-mail: samirkvaidya@yahoo.co.in, devsi.bantva@gmail.com Abstract: A distance two labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that | f(x)—f(y)| > 2 if d(x, y) = 1 and |f(x)—f(y)| => 1 if d(w,y) = 2. The L(2,1)-labeling number A(G) of G is the smallest number k such that G has an L(2, 1)-labeling with max{f(v) : v € V(G)} =k. Here we introduce a new graph family called generalized cactus and investigate the A-number for the same. Key Words: Channel assignment, interference, distance two labeling, block, cactus. AMS(2010): 05C78 §1. Introduction In a communication network, the main task is to assign a channel (non negative integer) to each TV or radio transmitters located at different places such that communication do not interfere. This problem is known as channel assignment problem which was introduced by Hale [4]. Usually, the interference between two transmitters is closely related with the geographic location of the transmitters. If we consider two level interference namely major and minor then two transmitters are very close if the interference is major while close if the interference is minor. Robert [7] proposed a variation of the channel assignment problem in which close transmitters must receive different channels and very close transmitters must receive channels that are at two apart. In a graph model of this problem, the transmitters are represented by the vertices of a graph; two vertices are very close if they are adjacent and close if they are at distance two apart in the graph. Motivated through this problem Griggs and Yeh [3] introduced L(2,1)- labeling which is defined as follows: Definition 1.1 A distance two labeling (or L(2,1)-labeling) of a graph G = (V(G), E(G)) is a function f from vertex set V(G) to the set of all nonnegative integers such that the following conditions are satisfied: 1Received March 14, 2012. Accepted June 25, 2012. 104 S.K. Vaidya and D.D.Bantva (1) |f(x) — f(y)| = 2 if d(a,y) = 1; (2) |f(z) —f(y)| = 1 if d(a,y) = 2. The span of f is defined as max{| f(u) — f(v) |: u,v € V(G)}. The A-number for a graph G, denoted by A(G) which is the minimum span of a distance two labeling for G. The L(2, 1)- labeling is studied in the past two decades by many researchers like Yeh [14]-[15], Georges and Mauro [2], Sakai [8], Chang and Kuo [1], Kuo and Yan [5], Lu et al. [6], Shao and Yeh [9], Wang [12], Vaidya et al. [10] and by Vaidya and Bantva [11]. We begin with finite, connected and undirected graph G = (V(G), E(G)) without loops and multiple edges. For the graph G, A denotes the maximum degree of the graph and N(v) denotes the neighborhood of v. Also in the discussion of distance two labeling [0, k] denotes the set of integers {0,1,--- ,&}. For all other standard terminology and notations we refer to West [13]. Now we will state some existing results for ready reference. Proposition 1.2({1]) ACUH) < A(G), for any subgraph H of a graph G. Proposition 1.3({14]) The A-number of a star Kya is A+1, where A is the maximum degree. Proposition 1.4((1]) If \(G) = A+1 then f(v) =0 or A+1 for any (G)-L(2, 1)-labeling f and any vertex v of maximum degree A. In this case, N{v] contains at most two vertices of degree A, for any vertex v € V(G). §2. Main Results The problem of labeling of trees with a condition at distance two remained the focus of many research papers as its A-number depends upon the maximum degree of a vertex. In [3], Griggs and Yeh proved that the \-number of any tree T with maximum degree A is either A+1 or A+2. They obtained the A-number by first-fit greedy algorithm. Later, trees are classified according to their \-numbers. The trees with A-number A + 1 are classified as class one otherwise they are of class two. Earlier it was conjectured that the classification problem is NP-complete but Chang and Kuo [1] presented a polynomial time classification algorithm. But even today the classification of trees of class two is an open problem. Motivated through this problem, we present here a graph family which is not a tree but its A-number is either A+ 1 or A+ 2 and it is a super graph of tree. A block of a graph G is a maximal connected subgraph of G that has no cut-vertex. An n-complete cactus is a simple graph whose all the blocks are isomorphic to K,,. We denote it by C(K,,). An n-complete k-regular cactus is an n-complete cactus in which each cut vertex is exactly in k blocks. We denote it by C(K,,(k)). The block which contains only one cut vertex is called leaf block and that cut vertex is known as leaf block cut vertex. We illustrate the definition by means of following example. Example 2.1 A 3-complete cactus C(K3) and 3-complete 3-regular cactus are shown in Fig.1 and Fig.2 respectively. Distance Two Labeling of Generalized Cacti 105 Fig.1 Fig.2 Theorem 2.2 Let C(Kn(k)) be an n-complete k-regular cactus with maximum degree A and k > 3. Then (C(K,(k))) ts either A+1 or A+2. Proof Let C(K,(k)) be an n-complete k-regular cactus with maximum degree A. The star Kya is a subgraph of C(K,,(k)) and hence A\(C(K,,(k))) > A+1. For upper bound, we apply the following Algorithm: Algorithm 2.3 The L(2,1)-labeling of given n-complete k-regular cactus. Input An n-complete k-regular cactus graph with maximum degree A. Idea Identify the vertices which are at distance one and two apart. Initialization Let vp be the vertex of degree A. Label the vertex vp by 0 and take S = {vo}- 106 S.K. Vaidya and D.D.Bantva Iteration Define f : V(G) — {0,1,2,...} as follows. Step 1 Find N(wvo). If N(vo) = {v1, ve ,..., va} then partition N(vo) into k sets Vi, Va, ... Ve such that for each i = 1,2,...,4 the graph induced by V; U {vo} forms a complete subgraph of C(K,,(k)). The definition of C(K,,(k)) itself confirms the existence of such partition with the characteristic that for i A 7, ue Vi, v € Vj, d(u,v) = 2. Step 2 Choose a vertex v; € N(vo) and define f(v1) = 2. Find a vertex v2 € N(vo) such that d(v1, v2) = 2 and define f(v2) = 3. Continue this process until all the vertices of N(vo) are labeled. Take S = {uo} U {uv € V(G)/f(v) is a label of v}. Step 3 For f(v;) = 7%. Find N(v;) and define f(v) = the smallest number from the set {0,1,2,---} — {i—1,7,7+ 1}, where v € N(v;) — S such that |f(u) — f(v)| > 2 if d(u,v) = 1 and |f(u) — f(v)| > 1 if d(u,v) =2. Denote S U {v € V(G)/f(v) is a label of v} = S?. Step 4 Continue this recursive process till S” = V(G), where S" = S"~1 U{v € V(G)/f(v) is a label of v}. Output max{f(v)/v €E V(G)} =A+42. Hence, A(C(Kn(k))) < A+2.Thus, (C(K,,(k))) is either A+1 or A+ 2. Example 2.4 In Fig.3, the £(2,1)-labeling of 3-complete 3-regular cactus C'(¢3(3)) is shown for which \(C(K3(3))) = A+2 = 8. Griggs and Yeh [3] have proved that: (1) A(P2) = 2; (2) A(P3) = A(P1) = 3, and (3) A(P,,) = 4, for n > 5. This can be verified by Proposition 1.4 and using our Algorithm 2.3. In fact, any path P, is 2-complete 2-regular cactus C'(K2(2)). Thus a single Algorithm will work to determine the \-number of path P,,. Using Algorithm 2.3 the L(2,1)-labeling of P2, P3, Py and Ps is demonstrated in Fig.4. Distance Two Labeling of Generalized Cacti 107 : a : : P» P4 poo 8 2 oo 8 P3 Ps Fig.4 Theorem 2.5 Let C(K,,) be an n-complete cactus with at least one cut verter which belongs to at least three blocks. Then \(C(Ky)) is either A+1 or A+2. Proof Let C(K,,) be the arbitrary an n-complete cactus with maximum degree A. The graph K1,q is a subgraph of C(K,,) and hence by Propositions 1.2 and 1.3 \(C(K,)) > A+1. Moreover C(K,,) is a subgraph of C(K,,(k)) (where k is -4,) and hence by Proposition 1.2 and n Theorem 2.2, \(C(K,)) < A+ 2. Thus, we proved that A(C(K,)) is either A+1 or A+2. Now as a corollary of above result it is easy to show that the A-number of any tree with maximum degree A is either A +1 or A+ 2. We also present some other graph families as a particular case of above graph families whose A-number is either A+ 1 or A+ 2. Corollary 2.6 Let T be a tree with maximum degree A > 2. Then X(T) is either A+ 1 or A +2. Proof Let T be a tree with maximum degree A > 2. If A = 2 then T is a path and problem is settled. But if A > 2 then A(T) > A+1 as Kj, is a subgraph of T. The upper bound of A-number is A + 2 according to Theorem 2.5 as any tree T' is a 2-complete cactus C(K2). Thus, we proved that X(T) is either A+1 or A +2. Example 2.7 In Fig.5, the L(2, 1)-labeling of tree T is shown which is 2-complete cactus C'(K2) with maximum degree A = 3 for which X(T) = A(C(K2)) = A+2 =5. Fig.5 108 S.K. Vaidya and D.D.Bantva Corollary 2.8 (Kin) =n+1. Proof The star Ky, is a 2-complete n-regular cactus. Then by Theorem 2.2, \(Ain) = n+1. Corollary 2.9 For the Friendship graph F,,, (Fn) = 2n+1. Proof The Friendship graph F;, is a 3-complete 2n-regular cactus. Then by Theorem 2.2, AM Fr) = 2n+1. Example 2.10 In Fig.6 and Fig.7, the L(2,1)-labeling of star Ky,4 and Friendship graph Fy are shown for which A-number is 5 and 9 respectively. 3 2 Fig.6 Fig.7 §3. Concluding Remarks We have achieved the A-number of an n-complete k-regular cactus. The A-numbers of some standard graphs determined earlier by Griggs and Yeh [3] can be obtained as particular cases of our results which is the salient feature of our investigations. Distance Two Labeling of Generalized Cacti 109 References 1 10 11 12 13 14 15 G.J.Chang and D.Kuo, The L(2,1)-labeling problem on graphs, SIAM J. Discrete Math., 9(2)(1996), 309-316. J.P.Georges and D.W.Mauro, Labeling trees with a condition at distsnce two, Discrete Math., 269(2003), 127-148. J.R.Griggs and R.K.Yeh, Labeling graphs with condition at distance 2, SIAM J. Discrete Math., 5(4)(1992), 586-591. W.K.Hale, Frequency assignment:Theory and applications, Proc. IEEE, 68(12)(1980), 1497-1514. D.Kuo and J.Yan, On L(2,1)-labelings of cartesian products of paths and cycles, Discrete Math., 283(2004), 137-144. C.Lu, L.Chen, M.Zhai, Extremal problems on consecutive L(2, 1)-labeling, Discrete Applied Math., 155(2007), 1302-1313. F.S.Robert, T-coloring of graphs: recent results and open problems, Discrete Math., 93(1991), 229-245. D.Sakai, Labeling chordal graphs: distance two condition, SIAM J. Discrete Math., 7(1)(1994), 133-140. Z.Shao and R.Yeh, The L(2,1)-labeling and operations of graphs, IEEE Transactions on Circuits and Systems-I, 52(3)(2005), 668-671. S.K.Vaidya, P.L.Vihol, N.A.Dani and D.D.Bantva, L(2,1)-labeling in the context of some graph operations, Journal of Mathematics Research, 2(3)(2010), 109-119. S.K.Vaidya and D.D.Bantva, Labeling cacti with a condition at distance two, Le Matem- atiche, 66(2011), 29-36. W.Wang, The L(2,1)-labeling of trees, Discrete Applied Math., 154(2006), 598-603. D.B.West, Introduction to Graph theory, Printice -Hall of India, 2001. R.K.Yeh, Labeling Graphs with a Condition at Distance Two, Ph.D.Thesis, Dept.of Math., University of South Carolina, Columbia,SC, 1990. R.K.Yeh, A survey on labeling graphs with a condition at distance two, Discrete Math., 306(2006), 1217-1231. As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. By Albert Einstein, an American theoretical physicist.. 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Mathematical Combinatorics (International Book Series) June, 2012 alg 9 Contents Neutrosophic Rings IT BY AGBOOLA A.A.A., ADELEKE E.O. and AKINLEYE S.A Non-Solvable Spaces of Linear Equation Systems BY LINFAN MAO Roman Domination in Complementary Prism Graphs BY B.CHALUVARAJU AND V.CHAITRA Enumeration of Rooted Nearly 2-Regualr Simple Planar Maps BY SHUDE LONG AND JUNLIANG CAI On Pathos Total Semitotal and Entire Total Block Graph of a Tree BY MUDDEBIHAL M. H. AND SYED BABAJAN On Folding of Groups BY MOHAMED ESMAIL BASHER....................52 On Set-Semigraceful Graphs BY ULLAS THOMAS AND SUNIL C. MATHEW On Generalized m-Power Matrices and Transformations BY SUHUA YE, YIZHI CHEN AND HUI LUO Perfect Domination Excellent Trees BY SHARADA B On (k,d)-Maximum Indexable Graphs and (k,d)-Maximum Arithmetic Graphs BY ZEYNAB KHOSHBAKHT Total Dominator Colorings in Paths BY A.VIJAYALEKSHMI Degree Splitting Graph on Graceful, Felicitous and Elegant Labeling BY P.SELVARAJU, P.BALAGANESAN, J-.RENUKA AND V.BALAJ Distance Two Labeling of Generalized Cacti BY S.K.VAIDYA AND D.D.BANTVA SEN SO VeI9S (30865 TedessMeM Ties ALGKe\)