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ISBN 978-1-59973-308-1 VOLUME 3, 2014 MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES) Edited By Linfan MAO THE MADIS OF CHINESE ACADEMY OF SCIENCES AND BEIJING UNIVERSITY OF CIVIL ENGINEERING AND ARCHITECTURE September, 2014 Vol.3, 2014 ISBN 978-1-59973-308-1 MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES) Edited By Linfan MAO The Madis of Chinese Academy of Sciences and Beijing University of Civil Engineering and Architecture September, 2014 Aims and Scope: The Mathematical Combinatorics (International Book Series) (ISBN 978-1-59973-308-1) 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; Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and map enumeration; Combinatorial designs; Combinatorial enumeration; Differential Geometry; Geometry on manifolds; 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. 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Subscription A subscription can be ordered by an email 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 (3nd) Editor-in-Chief Shaofei Du Linfan MAO Cepital Normal el verely, P.R.China ; : Email: dushf@mail.cnu.edu.cn Chinese Academy of Mathematics and System Science, P.R.China Baizhou He and Beijing University of Civil Engineering and Beijing University of Civil Engineering and Architecture, P-.R.China Architecture, P.R.China Email: hebaizhou@bucea.edu.cn Email: maolinfan@163.com Xiaodong Hu Chinese Academy of Mathematics and System Science, P.R.China Email: xdhu@amss.ac.cn Deputy Editor-in-Chief Guohua Song Beijing University of Civil Engineering and Yuangiu Huang Architecture, P.R.China Hunan Normal University, P.R.China Email: songguohua@bucea.edu.cn Email: hyqq@public.cs.hn.cn H.Iseri Editors Mansfield University, USA Email: hiseriQ@mnsfld.edu S.Bhattacharya : ; ; Xueliang Li Deakin University Nankai University, P.R.China Geelong Campus at Waurn Ponds . : Email: lxl@nankai.edu.cn Australia Email: Sukanto.Bhattacharya@Deakin.edu.au Guodong Liu Huizhou University Email: Igd@hzu.edu.cn Said Broumi Hassan IT University Mohammedia Hay El Baraka Ben M’sik Casablanca W.B.Vasantha Kandasamy B.P.7951 Morocco Indian Institute of Technology, India . , Email: vasantha@iitm.ac.in Junliang Cai Beijing Normal University, P-R.China Ion Patrascu Email: caijunliang@bnu.edu.cn Fratii Buzesti National College Craiova Romania Yanxun Chang Beijing Jiaotong University, P.R.China Han Ren Email: yxchang@center.njtu.edu.cn East China Normal University, P.R.China : : Email: hren@math.ecnu.edu.cn Jingan Cui Beijing University of Civil Engineering and Ovidiu-Ilie Sandru Architecture, P.R.China Politechnica University of Bucharest Email: cuijingan@bucea.edu.cn Romania li International Journal of Mathematical Combinatorics Mingyao Xu Y. Zhang Peking University, P.R.China Email: xumy@math.pku.edu.cn Department of Computer Science Georgia State University, Atlanta, USA Guiying Yan Chinese Academy of Mathematics and System Science, P.R.China Email: yanguiying@yahoo.com Famous Words: Too much experience is a dangerous thing. By Ooscar Wilde, A British dramatist. Math.Combin. Book Ser. Vol.3(2014), 01-84 Mathematics on Non-Mathematics — A Combinatorial Contribution Linfan MAO (Chinese Academy of Mathematics and System Science, Beijing 100190, P.R.China) E-mail: maolinfan@163.com Abstract: A classical system of mathematics is homogenous without contradictions. But it is a little ambiguous for modern mathematics, for instance, the Smarandache geometry. Let ¥ be a family of things such as those of particles or organizations. Then, how to hold its global behaviors or true face? Generally, ¥ is not a mathematical system in usual unless a set, ie., a system with contradictions. There are no mathematical subfields applicable. Indeed, the trend of mathematical developing in 20th century shows that a mathemati- cal system is more concise, its conclusion is more extended, but farther to the true face for its abandoned more characters of things. This effect implies an important step should be taken for mathematical development, i.e., turn the way to extending non-mathematics in classical to mathematics, which also be provided with the philosophy. All of us know there always exists a universal connection between things in #. Thus there is an underlying structure, i.e., a vertex-edge labeled graph G for things in ¥. Such a labeled graph G is invariant accompanied with ¥. The main purpose of this paper is to survey how to extend classical mathematical non-systems, such as those of algebraic systems with contradictions, algebraic or differential equations with contradictions, geometries with contradictions, and generally, classical mathematics systems with contradictions to mathematics by the under- lying structure G. All of these discussions show that a non-mathematics in classical is in fact a mathematics underlying a topological structure G, i.e., mathematical combinatorics, and contribute more to physics and other sciences. Key Words: Non-mathematics, topological graph, Smarandache system, non-solvable equation, CC conjecture, mathematical combinatorics. AMS(2010): 03A10,05C15,20A05, 34A26,35A01,51A05,51D20,53A35 §1. Introduction A thing is complex, and hybrid with other things sometimes. That is why it is difficult to know the true face of all things, included in “Name named is not the eternal Name; the unnamable is the eternally real and naming the origin of all things”, the first chapter of TAO TEH KING {9], a well-known Chinese book written by an ideologist, Lao Zi of China. In fact, all of things with 1Received February 16, 2014, Accepted August 8, 2014. 2 Linfan MAO universal laws acknowledged come from the six organs of mankind. Thus, the words “existence” and “non-existence” are knowledged by human, which maybe not implies the true existence or not in the universe. Thus the existence or not for a thing is invariant, independent on human knowledge. The boundedness of human beings brings about a unilateral knowledge for things in the world. Such as those shown in a famous proverb “the blind men with an elephant”. In this proverb, there are six blind men were asked to determine what an elephant looked like by feeling different parts of the elephant’s body. The man touched the elephant’s leg, tail, trunk, ear, belly or tusk respectively claims it’s like a pillar, a rope, a tree branch, a hand fan, a wall or a solid pipe, such as those shown in Fig.1 following. Each of them insisted on his own and not accepted others. They then entered into an endless argument. Fig.1 All of you are right! A wise man explains to them: why are you telling it differently is because each one of you touched the different part of the elephant. So, actually the elephant has all those features what you all said. Thus, the best result on an elephant for these blind men is Anelephant = {4 pillars} Ua rope} Ua tree branch} U {2 hand fans} Ua wall} Uta solid pipe} What is the meaning of this proverb for understanding things in the world? It lies in that the situation of human beings knowing things in the world is analogous to these blind men. Usually, a thing T is identified with its known characters ( or name ) at one time, and this process is advanced gradually by ours. For example, let w11, f12,--- , Un be its known and 1,7 > 1 unknown characters at time t. Then, the thing 7 is understood by 2 (Wim) Ul Uw (1) k>1 in logic and with an approximation T° = =U {ui} for T at time t. This also answered why difficult for human beings knowing a thing really, Mathematics on Non-Mathematics — A Combinatorial Contribution 3 Generally, let © be a finite or infinite set. A rule or a law on a set % is a mapping ux d---x } — E for some integers n. Then, a mathematical system is a pair (©;R), where a n R consists those of rules on © by logic providing all these resultants are still in U. Definition 1.1([28]-[30]) Let (%1;R1), (H2;R2), +--+, (Um; Rm) be m mathematical system, ~ m as m different two by two. A Smarandache multi-system X& is a union (J dX; with rules R = U Ri i=1 i=1 on ¥, denoted by (E:R). Consequently, the thing T is nothing else but a Smarandache multi-system (1.1). However, these characters 1%, k > 1 are unknown for one at time t. Thus, T » T° is only an approximation for its true face and it will never be ended in this way for knowing T, i.e., “Name named is not the eternal Name”, as Lao Zi said. But one’s life is limited by its nature. It is nearly impossible to find all characters vz, k > 1 identifying with thing 7. Thus one can only understands a thing T' relatively, namely find invariant characters J on vz, k > 1 independent on artificial frame of references. In fact, this notion is consistent with Erlangen Programme on developing geometry by Klein [10]: given a manifold and a group of transformations of the same, to investigate the configurations belonging to the manifold with regard to such properties as are not altered by the transformations of the group, also the fountainhead of General Relativity of Einstein [2]: any equation describing the law of physics should have the same form in all reference frame, which means that a universal law does not moves with the volition of human beings. Thus, an applicable mathematical theory for a thing T should be an invariant theory acting on by all automorphisms of the artificial frame of reference for thing T. All of us have known that things are inherently related, not isolated in philosophy, which implies that these is an underlying structure in characters p;, 1 <i<n for a thing T, namely, an inherited topological graph G. Such a graph G should be independent on the volition of human beings. Generally, a labeled graph G for a Smarandache multi-space is introduced following. Definition 1.2([21]) For any integer m > 1, let (5: ) be a Smarandache multi-system con- sisting of m mathematical systems (X11; 71), (H2;R2), +++, (mj Rm). An inherited topological structure GS] of (5:8) is a topological vertex-edge labeled graph defined following: V(G([S]) = {X1, Daye Um}, E(G[S]) = {(5,E;)|Xi QE; 40, 1<i4 5 <m} with labeling for integers 1<iAg<m. However, classical combinatorics paid attentions mainly on techniques for catering the need of other sciences, particularly, the computer science and children games by artificially giving up individual characters on each system (X,7#). For applying more it to other branch sciences initiatively, a good idea is pullback these individual characters on combinatorial objects again, 4 Linfan MAO ignored by the classical combinatorics, and back to the true face of things, i.e., an interesting conjecture on mathematics following: Conjecture 1.3(CC Conjecture, [15],[19]) A mathematics can be reconstructed from or turned into combinatorization. Certainly, this conjecture is true in philosophy. So it is in fact a combinatorial notion on developing mathematical sciences. Thus: (1) One can combine different branches into a new theory and this process ended until it has been done for all mathematical sciences, for instance, topological groups and Lie groups. (2) One can selects finite combinatorial rulers and axioms to reconstruct or make general- izations for classical mathematics, for instance, complexes and surfaces. From its formulated, the CC conjecture brings about a new way for developing mathematics , and it has affected on mathematics more and more. For example, it contributed to groups, rings and modules ([11]-[14]), topology ({23]-[24]), geometry ([16]) and theoretical physics ([17]- [18]), particularly, these 3 monographs [19]-[21] motivated by this notion. A mathematical non-system is such a system with contradictions. Formally, let & be mathematical rules on a set ©. A pair (U;@) is non-mathematics if there is at least one ruler R € & validated and invalided on © simultaneously. Notice that a multi-system defined in Definition 1.1 is in fact a system with contradictions in the classical view, but it is cooperated with logic by Definition 1.2. Thus, it lights up the hope of transferring a system with contradictions to mathematics, consistent with logic by combinatorial notion. The main purpose of this paper is to show how to transfer a mathematical non-system, such as those of non-algebra, non-group, non-ring, non-solvable algebraic equations, non-solvable or- dinary differential equations, non-solvable partial differential equations and non-Euclidean ge- ometry, mixed geometry, differential non- Euclidean geometry, ---, etc. classical mathematics systems with contradictions to mathematics underlying a topological structure G, i.e., math- ematical combinatorics. All of these discussions show that a mathematical non-system is a mathematical system inherited a non-trivial topological graph, respect to that of the classical underlying a trivial K, or Kz. Applications of these non-mathematic systems to theoretical physics, such as those of gravitational field, infectious disease control, circulating economical field can be also found in this paper. All terminologies and notations in this paper are standard. For those not mentioned here, we follow [1] and [19] for algebraic systems, [5] and [6] for algebraic invariant theory, [3] and [32] for differential equations, [4], [8] and [21] for topology and topological graphs and [20], [28]-[31] for Smarandache systems. §2. Algebraic Systems Notice that the graph constructed in Definition 1.2 is in fact on sets 4;, 1 <i < m with relations on their intersections. Such combinatorial invariants are suitable for algebraic systems. All operations o: & x M — PM on a set & considered in this section are closed and single valued, i.e., ao b is uniquely determined in &/, and it is said to be Abelian if aob = boa for Mathematics on Non-Mathematics — A Combinatorial Contribution 5 Va,beE of. 2.1 Non-Algebraic Systems An algebraic system is a pair (#;R) holds with aob € & for Va,b EE MW andoe R. A non-algebraic system =(/;R) on an algebraic system (&;R) is AS~!: there maybe exist an operation o € R, elements a,b € A with ao b undetermined. Similarly to classical algebra, an isomorphism on —(&/;R) is such a mapping on & that for Vo € R, h(aob) = h(a) 0 h(b) holds for Va,b € &@ providing ao b is defined in =(#;R) and h(a) = h(b) if and only if a = b. Not loss of generality, let o € R be a chosen operation. Then, there exist closed subsets @, i > 1 of &. For instance, (a)° = {a,aoa,acaoa,::: ,€0G0+:-0a,-+-} — Ee k is a closed subset of & for Va € &. Thus, there exists a decomposition #°, Z>,---, &? of & such that aob € &° for Va,b € &° for integers 1 <i <n. Define a topological graph G[7(.#/; 0)] following: VGH 2)]) = {0 Bs Gah E(G[-(#;0)]) = {(@°, F}) if AEP AOL Si Aj <n} with labels L: #2 EV(G(P;0))) > L(@P) = &, L: (2,2) € E(G[>(a#;0)]) > #2 GEA for integers 1<ifAj <n. For example, let off = {a,b,c}, off = {a,d,f}, of? = {ede}, 2 = {a,e,f} and ae = {d,e, f}. Calculation shows that #°()@ = {a}, HI) MP = {c}, ML = {a}, de(\ade = 0, delle = {d}, Ala? = {a}, AE(\se = {af (\oe = {eh, As (|e = {d,e} and #P(\#e = {e,f}. Then, the labeled graph G[=(#;0)] is shown in Fig.2. 6 Linfan MAO Let h : & — & be an isomorphism on -(.#;0). Then Va,b € &%°), h(a) o h(b) = h(a ob) € h(A?) and h(A?) 1) h(A$) = h(AP?() AZ) = 0 if and only if A?) A$ = 0 for integers 1<i#j<n. Whence, if G"[=(#;0)| defined by V(G"[>(#;0)]) = {h( AP), R(@E), > RCM): E(G"[>(;0)]) = {(h(G?), h( A?) if WAP) h(@) AOL Si AG <n} with labels L': RA?) € V(G"[-(#;0)]) > L(h(e@?)) = h(@?), L': (h(x), (7) € E(G"[>(;0)]) > hf?) (| h@) for integers 1 <i4# gj < n. Thus h: & — & induces an isomorphism of graph h* : G[>(#;0)] > G"[-(#;0)]. We therefore get the following result. Theorem 2.1 A non-algebraic system =(/;0) in type AS~* inherits an invariant G[=(@; 0)| of labeled graph. Let CHAR) = U Gh(%9)] o€R be a topological graph on 7=(</;R). Theorem 2.1 naturally leads to the conclusion for non- algebraic system —(.7; R) following. Theorem 2.2 A non-algebraic system —(@;R) in type AS“! inherits an invariant G[>(@; R)] of topological graph. Similarly, we can also discuss algebraic non-associative systems, algebraic non-Abelian sys- tems and find inherited invariants G[=(./; 0)] of graphs. Usually, we adopt different notations for operations in R, which consists of a multi-system (./;R). For example, R = {+,-} in an algebraic field (R;+,-). If we view the operation + is the same as -, throw out 0-a, a-0 and 1+a, a+1 for Va € Rin R, then (R;+,-) comes to be a non-algebraic system (R;-) with topological graph G[R;-] shown in Fig.3. R\ {1} aE R\ {0} Fig.3 2.2 Non-Groups A group is an associative system (Y; 0) holds with identity and inverse elements for all elements in Y. Thus, for a,b,c € GY, (aob)oc=ao (boc), dlg € such that lyoa=aolg =a and for Va € Y, Ja~! € AY such that aca~' =1g. A non-group ~(Y; 0) on a group (Y;0) is an algebraic system in 3 types following: Mathematics on Non-Mathematics — A Combinatorial Contribution 7 AG_': there maybe exist a1,b1,c1 and ag, be,co €Y such that (a, 0b,) oc, = ay 0 (b1 0c1) but (az 0 bz) 0 cg # az 0 (be 02), also holds with identity 1g and inverse element a~} for all elements ina € G. AG: there maybe exist distinct ly, 1y € G such that ajo lg = lgoa, = a, and a2 0 ly = 140 a2 = ag for a, F az EY, also holds with associative and inverse elements at on lg and ly for Va EG. AG;?: there maybe exist distinct inverse elements a~',a~' for a € GY, also holds with associative and identity elements. Notice that (ao a) oa = ao (aoa) always holds with a € Y in an algebraic system. Thus there exists a decomposition 4,%,---,G, of Y such that (Y;;0) is a group for integers 1<i<n for Type AG{’. Type AG; is true only if lyoly # ly and # lly. Thus ly and 1 are local, not a global identity on Y. Define Glg)={aEe¥ ifacly=lyoa=a}. Then Y(ly) 4 Y(1y) if ly 4 14. Denoted by I(Y) the set of all local identities on Y. Then Gilg), ly € I(Y) is a decomposition of Y such that (¥(1g);0°) is a group for Vlg € I(¥Y). Type AG;’ is true only if there are distinct local identities ly on Y. Denoted by I(Y) the set of all local identities on Y. We can similarly find a decomposition of Y with group (Y(1g);0) holds for Vlg € I(¥) in this type. Thus, for a non-group —(; 0) of AG] '-AG3 ', we can always find groups (4; 0), (;0),-* , (Gn; 0) for an integer n > 1 with Y¥ = LU Y. Particularly, if (Y; 0) is itself a group, then such i=1 a decomposition is clearly exists by its subgroups. Define a topological graph G[=(G; °)] following: V(GI(F;9))) = {A, 4° Gnd EGG; 0)]) ={GiGH) #fAl)G 401 < 1,45 <n} with labels L: YE V(GIA(G;0)]) — L(G) =, L: (GG) € E(G(G;0)]) + G [|G for integers 1<i Aj <n. For example, let @ = (a, 3), @ = (a,7,9), #@ = (8,7), G% = (G,6,0) be 4 free Abelian groups witha #4 6 4 y #6 # O. Calculation shows that (1% = (a), ANS = (), G3(1G%. = (6), ANG = (GB) and @lI|% = (0). Then, the topological graph G[A(¥Y;0)] is shown in Fig.4. 8 Linfan MAO G, (a) Gp Ge (6) G3 Fig.4 For an isomorphism g : Y — Y on —~(G;0), it naturally induces a 1-1 mapping g* : V(GIF(Y;°)]) — V(G[A(Y; °)]) such that each g*(Y;) is also a group and g*(%) (19° (G;) #9 if and only if Y(|Y; # 0 for integers 1 < i 4 j <n. Thus g induced an isomorphism g* of graph from G[=(G; 0)] to g*(G[A(G; °)]), which implies a conclusion following. Theorem 2.3. A non-group —(Y;0) in type AG] '-AG3' inherits an invariant G[-(Y;0)] of topological graph. Similarly, we can discuss more non-groups with some special properties, such as those of non-Abelian group, non-solvable group, non-nilpotent group and find inherited invariants G|7>(G;0)]. Notice that({19]) any group Y can be decomposed into disjoint classes CH), C(A2),--- ,C(H,) of conjugate subgroups, particularly, disjoint classes Z(a1), Z(a2), +--+ , Z(az) of centralizers with |C(H;)| = |¥ : Ng(Ai)|, |Z(a;)| = |¥ : Zg(a;)|, 1<i<s,1<j <land |C(Ai)| + |C(A2)|+-+-+|CCHs)| = IYI, |2(a1)| + |2(a2)| +++» +|Z(ai)| = |¥|, where Ng (H), Z(a) denote respectively the normalizer of subgroup H and centralizer of element a in group ¢Y. This fact enables one furthermore to construct topological structures of non-groups with special classes of groups following: Replace a verter GY, by s; (or 1,) isolated vertices labeled with C(H1), C(H2),--- , C(Hs,) (or Z(a,), Z(az), «+ ,Z(ai,)) in G[A(Y;0)] and denoted the resultant by G[A(G;0)]. We then get results following on non-groups with special topological structures by Theorem 2.3. Theorem 2.4 A non-group —(Y;0) in type AG]'-AG3' inherits an invariant GIG; o)| of topological graph labeled with conjugate classes of subgroups on its vertices. Theorem 2.5 A non-group —(Y;0) in type AG]'-AG3' inherits an invariant GIG; o)| of topological graph labeled with Abelian subgroups, particularly, with centralizers of elements in GY on its vertices. Particularly, for a group the following is a readily conclusion of Theorems 2.4 and 2.5. Corollary 2.6 A group (Y;0) inherits an invariant Gg; 0] of topological graph labeled with ny conjugate classes of subgroups (or centralizers) on its vertices, with E(G[Y;0]) = Mathematics on Non-Mathematics — A Combinatorial Contribution 9 2.3 Non-Rings A ring is an associative algebraic system (R;+,0) on 2 binary operations “+”, “o”, hold with an Abelian group (R; +) and for Vz,y,z € R, ro(y+z) =xoyt+aroz and (x+y)oz =xoz+yoz. Denote the identity by 0,, the inverse of a by —a in (R;+). A non-ring =(R;+,0°) on a ring Rag 9 (R;+,0) is an algebraic system on operations “+”, “o” in 5 types following: AR{?: there maybe exist a,b € R such that a+b #6+<a, but hold with the associative in (R;0) and a group (R;+); AR;!: there maybe exist a1, b1,c1 and dg, be, c2 € R such that (a; 0b)) oc, = a, 0(b1 0¢1), (az 0 bz) 0 co F az O (b2 OC), but holds with an Abelian group (R; +). AR;!: there maybe exist a1, bi, c1 and ag, bg, co € R such that (a1 +b1)+c1 = ai +(b1+c1), (az + be) + cg 4 a2 + (bo + C2), but holds with (ao b)oc = ao (boc), identity 04 and —a in (R; +) for Va,b,cE R. ARj;': there maybe exist distinct 04,04 € R such that a+ 0; = 0; +a = a and b+0,.=0..+b=6 fora #b€ R, but holds with the associative in (R;+), (R; 0) and inverse elements —a on 0,, 04. in (R;+) for Va € R. AR;?!: there maybe exist distinct inverse elements —a,—a for a € R in (R;+), but holds with the associative in (R;+), (R;0o) and identity elements in (R; +). Notice that (a+a)+a=a+(a+a),a+a=a+a and aoa = ao always hold in non-ring -=(R;+,0°). Whence, for Types AR," and AR 1 there exists a decomposition R1, R2,--- ,Rn of R such that a+ b = b+a and (a0b)oc=a0 (boc) if a,b,c € Ri, ie., each (Ri; +, 0°) is a ring for integers 1 <i <n. A similar discussion for Types AG; *-AG3" in Section 2.2 also shows such a decomposition (R;;+,°), 1 <i <n of subrings exists for Types 3— 5. Define a topological graph G[=(R; +4, 0)] by V(G[>(R; +,0)]) = {Ri, Ra,--+ Rn}; E(G[-(R; +,°)]) = {(Ri, Rj) if Rif] Ry #01 <4, 45 <n} with labels DL: R€ V(G[7(R; +,0)]) = L(R;) = Rj, L: (Ri, R;) € E(G[A(R; +,°)]) > Ri( Rj for integers 1<iAj <n. Then, such a topological graph G[=(R; +, 0)] is also an invariant under isomorphic actions on -(R;+,0). Thus, Theorem 2.7 A non-ring =(R;+,0) in types AR; ‘-AR;5' inherits an invariant G[-(R; +4, °)] of topological graph. Furthermore, we can consider non-associative ring, non-integral domain, non-division ring, skew non-field or non-field, ---, etc. and find inherited invariants G[=(R;+,0)] of graphs. For example, a non-field =(F';+,0) on a field (F;+,0) is an algebraic system on operations “+”, Oa o” in 8 types following: 10 Linfan MAO AF7?: there maybe exist aj, b1,c, and ag, b2, co € F such that (a1 0b) oc, = a1 0 (by 0c1), (az 0b) 0cg 4 ag 0 (b2 C2), but holds with an Abelian group (F; +), identity 1., a~+ fora € F in (F;0). AF;’: there maybe exist a1, b1,¢1 and ag, be, co € F such that (a; +61)+¢e1 = ai +(bi+c1), (a2 + bg) + cg F ag + (b2 + ce), but holds with an Abelian group (F;0), identity 14, —a for a € F in (F;+). AF;!: there maybe exist a,b € F such that aob #4 boa, but hold with an Abelian group (F; +), a group (F; 0); AF;’: there maybe exist a,b € F such that a+b #b-+<a, but hold with a group (F; +), an Abelian group (F; 0); AF;': there maybe exist distinct 04,0/, € F such that a+ 0, =04+a=aandb+04, = 04. +b=b fora #4 b € F, but holds with the associative, inverse elements —a on 04, 04, in (F;+) for Va € F, an Abelian group (F; 0°); AF;": there maybe exist distinct 1.,15 € F such that aol, =1l,ca=aand bol} = 1,ob=b fora be F, but holds with the associative, inverse elements a~' on lo, 14, in (F;0) for Va € F, an Abelian group (F; +); AF? '; there maybe exist distinct inverse elements —a,—a for a € F in (F;+), but holds with the associative, identity elements in (F;+), an Abelian group (F; 0). AF, ': there maybe exist distinct inverse elements a~', a7! for a € F in (F;0), but holds with the associative, identity elements in (F;0), an Abelian group (F; +). Similarly, we can show that there exists a decomposition (F;;+,0), 1 <i<n of fields for non-fields =(F;+,°) in Types AF; ‘-AFg* and find an invariant G[-(F;+,0)] of graph. Theorem 2.8 A non-ring =(F;+,0) in types AF, '-AF* inherits an invariant G[-(F; +, °)] of topological graph. 2.4 Algebraic Combinatorics All of previous discussions with results in Sections 2.1-2.3 lead to a conclusion alluded in philosophy that a non-algebraic system 7(</;R) constraint with property can be decomposed into algebraic systems with the same constraints, and inherits an invariant G[7=(@;R)| of topological graph labeled with those of algebraic systems, i.e., algebraic combinatorics, which is in accordance with the notion for developing geometry that of Klein’s. Thus, a more applicable approach for developing algebra is including non-algebra to algebra by consider various non-algebraic systems constraint with property, but such a process will never be ended if we do not firstly determine all algebraic systems. Even though, a more feasible approach is by its inverse, i.e., algebraic G-systems following: Definition 2.9 Let (4;R1),(2%;R2),--+,(GriRn) be algebraic systems. An algebraic G- system is a topological graph G with labeling L: v € V(G) > Li(v) € {GH,&h,--- , GH} and L: (u,v) € E(G) > L(u)() Lv) with L(u) (1) L(v) 4 0, denoted by Gla, R], where fH = U & i=1 {= Mathematics on Non-Mathematics — A Combinatorial Contribution 11 and R = U Ri. i=1 Clearly, if G[.a/, R] is prescribed, these algebraic systems (4; R1), (4; R2), ++: , (Haj Rn) with intersections are determined. Problem 2.10 Characterize algebraic G-systems G[&@/,R], such as those of G-groups, G-rings, integral G-domain, skew G-fields, G'-fields, ---, etc., or their combination G — {groups, rings}, G — {groups, integral domains}, G — {groups, fields}, G — {rings, fields} ---. Particularly, characterize these G-algebraic systems for complete graphs G = Ko, K3,K4, path P3, P, or circuit C4 of order< 4. In this perspective, classical algebraic systems are nothing else but mostly algebraic Ky- systems, also a few algebraic K2-systems. For example, a field (F';+,-) is in fact a Ko-group prescribed by Fig.3. §3. Algebraic Equations All equations discussed in this paper are independent, maybe contain one or several unknowns, not an impossible equality in algebra, for instance 2*+¥** = 0. 3.1 Geometry on Non-Solvable Equations Let (LES), (LES?) be two systems of linear equations following: r=y r+y=1 =-y rt+ty=4 (LES%) (LES?) v= 2y g-y=l1 v= —2y xr—y=A4 Clearly, the system (LES}) is solvable with x = 0,y = 0 but (LES?) is non-solvable because x+y =1 is contradicts to that of x+y = 4 and so forr—y=1tox—y=A4. Even so, is the system (LES?) meaningless in the world? Similarly, is only the solution x = 0, y = 0 of system (LES}) important to one? Certainly NOT! This view can be readily come into being by all figures on R? of these equations shown in Fig.5. Thus, if we denote by Ly = {(2,y) € R’la = y} Li ={(2,y) € Plz +y = 1} Lo = {(2,y) € R?|x = —y} =A Ly = {(z,y) € R’lz + y = 4} Ls = {(x,y) € R?|a = 2y} Lk = {(a,y) €R|a—-y=1} | La = {(2,y) € R?|x = —2y} Li, = {(2,y) € R’|x — y = 4} 12 Linfan MAO (LES}) (LES?) Fig.5 the global behavior of (LES}), (LES?) are lines L1 — La, lines Li — L’4 on R? and Lif )Lof \Ls(\L4 = {(0,0)} but L415) L404 = 0. Generally, let fi(t1,2,+++,%n) =0 (ESm) fo(a1, v2, ro ) fim(@1, £2,°° 7 , In) =0 be a system of algebraic equations in Euclidean space R” for integers m,n > 1 with non-empty point set Sr, C R” such that fi(r1,22,--- ,¢n) = 0 for (a1, 2,--- ,In) © Sp, 1 St <m. Clearly, the system (£S,,) is non-solvable or not dependent on ()Sr=0 or #90. i=1 Conversely, let Y be a geometrical space consisting of m parts G%,,%,--- ,Ym in R”, where, each Y, is determined by a system of algebraic equations f(a, L2,°°° ln) — 0 fa, T2,°°° yn) = 0 fl (x1, 22, yo ,En) =0 Then, the system of equations Mathematics on Non-Mathematics — A Combinatorial Contribution 13 is non-solvable or not dependent on m (\G=0 or 40. i=1 Thus we obtain the following result. Theorem 3.1 The geometrical figure of equation system (ES) is a space Y consisting of m parts GY, determined by equation fi(%1,22,--+,%n) =0, 1 <i<m in (ES,,), and is non- solvable if a G, =. Conversely, if a geometrical space Y consisting of m parts,4Y,,G, ++: ,Gm; each of thenias determined by a system of algebraic equations in R”, then all of these equations m consist a system (ES,,), which is non-solvable or not dependent on (| Y; = or not. i=1 For example, let G be a planar graph with vertices vj, v2, v3, v4 and edges v1 V2, U1 U3, V2U3, U3U4, U4U1, Shown in Fig.6. Fig.6 Then, a non-solvable system of equations with figure G on R? consists of nS. y=8 (LEs)4 x =12 y=2 3a + 5y = 46. Thus G is an underlying graph of non-solvable system (LEs). Definition 3.2 Let (ES»,) be a solvable system of m; equations fl (a1, 22, -°- tre) =0 flan, 22,--- ti) =0 14 Linfan MAO with a solution space S'r in R” for integers 1<i<m. A topological graph G|ES,] is defined by V(G|ESm]) = {S ta, 1<i<m}; E(GIESm]) = {(Spa, Spur) if Spa (Span #9, 1 <i 4G <m} with labels LD: Spa € V(G[ES;,)]) > L(S pt) = Sra, L: (Spa, Spt) € E(G[ES;,)]|) => Sta) () Siu for integers 1<i4#j<m. Applying Theorem 3.1, a conclusion following can be readily obtained. Theorem 3.3 A system (ESm) consisting of equations in (ES, ), 1 <i <m is solvable if and only if GIESm] ~ Km withDASC 1) Spy. Otherwise, non-solvable, i.e., G[ESm] # Km, or i=1 G[ESm] ~ Km but (1) Sym = 9. i=l Let T: (@1,%2,°++ ,4n) > (a},25,---,2/,) be linear transformation determined by an invertible matrix [ajj],,...,> Le, ©; = i121 +aj2%2+-+-+Gjn%n, 1 <i < nand let T (Sim) = Sm for integers 1 <k <™m. Clearly, T: {Spm, 1<i<m}— {Seta 1<i<m} and Srtal Siu AO if and only if Spa (| Spun #0 for integers 1 <i #7 <_m. Consequently, if T: (ES) <— (‘ESy), then G[ES,,] ~ GES]. Thus T induces an isomorphism T* of graph from G/ES,,] to G[/ES,,], which implies the following result: Theorem 3.4 A system (ES,,) of equations f;(Z) = 0,1 <i < m inherits an invariant G[ES | under the action of invertible linear transformations on R”. Theorem 3.4 enables one to introduce a definition following for algebraic system (E'S,,,) of equations, which expands the scope of algebraic equations. Definition 3.5 If G[ES,,] is the topological graph of system (ESi,) consisting of equations in (ESm,) for integers 1 <4 <m, introduced in Definition 3.2, then G[ES;,] is called a G-solution of system (E'S'y,). Thus, for developing the theory of algebraic equations, a central problem in front of one should be: Problem 3.6 For an equation system (ES), determine its G-solution G[ESn]. For example, the solvable system (£S,,,) in classical algebra is nothing else but a Ky,- solution with (] Sia A 0, as claimed in Theorem 3.3. The readers are refereed to references i=l [22] or [26] for more results on non-solvable equations. Mathematics on Non-Mathematics — A Combinatorial Contribution 15 3.2 Homogenous Equations A system (E'S,,) is homogenous if each of its equations f;(%0,%1,--:,%n), 1 < i < m is homogenous, i.e., fi(Avo, AX1, eee jAtn) = r f;(z0, 21, ed i) for a constant , denoted by (hES,,). For such a system, there are always existing a Ky,- m solution with {x; = 0, 0 <i<n}C [) Sp and each fi(%o,21,--- ,@n) = 0 passes through i=1 O = (0,0,--- ,0) in R”. Clearly, an invertible linear transformation T action on such a K,,- — n+1 solution is also a Ky,-solution. However, there are meaningless for such a K,,-solution in projective space P” because O ¢ P”. Thus, new invariants for such systems under projective transformations (7p, 24,--- ,2},) = [i] (41) x(n41) (0, %1,"** ,Xn) should be found, where [ajj] ) is invertible. In R’, (n+1)x(n+41 two lines P(x, y), Q(x, y) are parallel if they are not intersect. But in P?, this parallelism will never appears because the Bézout’s theorem claims that any two curves P(x,y,z), Q(z,y, 2) of degrees m,n without common components intersect precisely in mn points. However, de- noted by I(P,Q) the set of intersections of homogenous polynomials P(%) with Q(%) with E = (Xo, 21,°+: , Xn). The parallelism in R” can be extended to P” following, which enables one to find invariants on systems homogenous equations. Definition 3.7 Let P(Z),Q(Z) be two complex homogenous polynomials of degree d with & = (%0,%1,°++,2n). They are said to be parallel, denoted by P || Q ifd > 1 and there are constants a,b,--+,¢ (not all zero) such that for VE € I(P,Q), avo + ba, +--+ + cap, = 0, te, all intersections of P(Z) with Q(Z) appear at a hyperplane on P’C, or d = 1 with all intersections at the infinite x, =0. Otherwise, P(Z) are not parallel to Q(Z), denoted by P | Q. Definition 3.8 Let P,(Z) = 0, Po(%) =0,--- , Pm(¥) = 0 be homogenous equations in (RES). Define a topological graph G[IhES,,| in P” by V(G[hESm]) = {Pi (2), Po(Z), +++, Pm(®)}; E(G|RESm]) = {(Pi(®), Pi(®))|Pi WP), 1S t,7 Sm} with a labeling L: Pi(@)—> PZ), (Pi(®), P)(®)) — 1B, Pj), where 1 <i fj <m. For any system (hE‘S,,) of homogenous equations, G[hE'S,,] is an indeed invariant under the action of invertible linear transformations T on P”. By definition in [6], a covariant C(ag, Z) on homogenous polynomials P(%) is a polynomial function of coefficients ag and variables 7. We furthermore find a topological invariant on covariants following. Theorem 3.9 Let (hES,) be a system consisting of covariants C;(ag,%) on homogenous polynomials P;(Z) for integers 1<i<m. Then, the graph G[RES] is a covariant under the action of invertible linear transformations T, i.e., for VC;(az,) € (ESm), there is Cy (ag, Z) € (ESm) with Cy (az, 7’) = APC; (az, Z) 16 Linfan MAO holds for integers 1<1i<m, where p is a constant and A is the determinant of T. Proof Let GT[hES,,| be the topological graph on transformed system T(hES,,) defined in Definition 3.8. We show that the invertible linear transformation TJ naturally induces an isomorphism between graphs G[hES,,] and G7 [hES;,]. In fact, T naturally induces a mapping T* : G[hESm] — GT[hES,,] on P". Clearly, T* : V(G[RESm]) —~ V(GT[hESm]) is 1-1, also onto by definition. In projective space P”, a line is transferred to a line by an invertible linear transformation. Therefore, C7 || C7 in T(ES,,) if and only if C, || Cy in (hRESin), which implies that (C7,C?) € E(GT[ES,,]) if and only if (Cu,C.) € E(G[hES,,]). Thus, G[hES im] ~ GT [RES in] with an isomorphism T* of graph. Notice that I (C7,C7) = T (I(Cy,Cy)) for V(Cu, Cy) € E(G[RES;,]). Consequently, the induced mapping T*: V(G[hESm]) > V(G" [RESm]), E(G[hES]) — E(G7 [hESm]) is commutative with that of labeling LD, i.e, T* oD = LoT*. Thus, T* is an isomorphism from topological graph G[hES\,] to G7 [hE Sn]. Particularly, let p = 0, ie., (ES;,) consisting of homogenous polynomials P;(%), P2(Z), - , P(X) in Theorem 3.9. Then we get a result on systems of homogenous equations following. Corollary 3.10 A system (hES) of homogenous equations f;(Z) = 0,1 <i <m inherits an invariant G[|RES;,] under the action of invertible linear transformations on P”. Thus, for homogenous equation systems (hE'S;,), the G-solution in Problem 3.6 should be substituted by G[AES,,]-solution. §4. Differential Equations 4.1 Non-Solvable Ordinary Differential Equations For integers m, n > 1, let X=F,(X),1<i<m (DES) , dX ee be a differential equation system with continuous fF, :R” — R”, X = ere such that F;(0) = 0, particularly, let X= A,X,--- ,X = AgX,--- , X = Am X (LDES}) be a linear ordinary differential equation system of first order with , 3 é : dx, dx dxn, X= ++ dy)§ = (—, —,--- , (a a »v ) ( dt ’ dt ’ ’ dt ) and g®™ + al) g(m—1) +... ft ally = 0 n 0) (n- 0 a” 4 aXler—D 4...4 ale =0 pes Mathematics on Non-Mathematics — A Combinatorial Contribution 17 a linear differential equation system of order n with [k] [A [k] Qi Ag Ain r1(t) k k k ee oe eel Gt <td Seas: anh tn (t) Lae 0<k<m, 1<i,j <n are numbers. Such a system (DES},) or (LDES}.) (or (LDE®,)) are called non-solvable if there are no function X(t) (or x(t)) hold with (DES!) or (LDES},) (or (LDE®.)) unless constants. For example, the following differential equation system where, «) = ae all a — 3% + 2c =0 ( é —5é + 6x =0 ( i — Te + 122 = 0 (HDR ge noes ( & — 94 + 202 = 0 ( @—1lé+30r=0 ( ( &-—7Tz+ 6x =0 is a non-solvable system. According to theory of ordinary differential equations ([32]), any linear differential equation system (LDES}) of first order in (LDES},) or any differential equation (LDE?) of order n with complex coefficients in (L DE") are solvable with a solution basis Z = { B;(t)| 1 <i<n} such that all general solutions are linear generated by elements in &. Denoted the solution basis of systems (DES},) or (LDES},) (or (LDE™)) of ordinary dif- ferential equations by 41, F2,--- , Am and define a topological graph G[DES},] or G[LDES}] (or G[LDE?]) in R” by V(G[DES,,]) = V(G[LDES,,]) = V(G[LDE},]) = {#, Ba,» , Bm}; E(G|DES),]) = E(G[LDES},]) = E(G|LDE®)) = {(Bi,B;) if Bl )B #9, 1<i,7<m} with a labeling L: BB, (Bi, B;) > Bil \B; for 1 <i#j<m. Let T be a linear transformation on R” determined by an invertible matrix [a,;| Let nxn’ T: {B, 1<i<m} 3 {BH, 1<i<m}. It is clear that &; is the solution basis of the ith transformed equation in (DES},) or (LDES},) (or (LDEy,)), and B()\B; A 0 if and only if (| 4; # 0. Thus T naturally induces an isomorphism 7T* of graph with T* o L = Lo T™ on labeling L. 18 Linfan MAO Theorem 4.1 A system (DES!) or (LDES}) (or (LDE™,)) of ordinary differential equations inherits an invariant G[DES},] or G[LDES}] (or G[LDE™]) under the action of invertible linear transformations on R”. Clearly, if the topological graph G[DES}] or G[LDES},] (or G[LDE”,]) are determined, the global behavior of solutions of systems (DES},) or (LDES},) (or (LDE™)) in R” are readily known. Such graphs are called respectively G[DES},]-solution or G[LDES}\,|-solution (or G[LDE™ |-solution) of systems of (DES?) or (LDES}) (or (LDE”,)). Thus, for developing ordinary differential equation theory, an interesting problem should be: Problem 4.2 For a system of (DES},) (or (LDES},), or (LDE”)) of ordinary differential equations, determine its G[DES},]-solution ( or G[LDES},]-solution, or G[LDE® |-solution). For example, the topological graph G[L DE%] of system (LZ DE@?) of linear differential equa- tion of order 2 in previous is shown in Fig.7. {e"} {e*"} {et et} {e3t, ety Jee, er} {er} tert er) Fig.7 4.2 Non-Solvable Partial Differential Equations Let [1, L2,--- , Lm be m partial differential operators of first order (linear or non-linear) with ” O Le =) ani 1l<k<m. i=1 Then the system of partial differential equations Ljlu(a1, 22,°+* ,&n)| = hi, 1l<i<m, (PDES,,) or the Cauchy problem a (PDESS) u(r1,%2,°°° nist. NOs l<i<m is non-solvable if there are no function u(a1,--- ,¢p) on a domain D C R” with (PDES,,) or (PDESC) holds, where hj, 1 <i < mand w 1 <i< ™m are all continuous functions on DCR". Mathematics on Non-Mathematics — A Combinatorial Contribution 19 Clearly, the ith partial differential equation is solvable [3]. Denoted by S? the solution of ith equation in (PDES,,) or (DEPS&). Then the system (PDES,,) or (DEPS®) of partial differential equations is solvable only if () S? 4 @. Because u : R" — R” is differentiable, so i=1 the (PDES\,,) or (DEPS®) is solvable only if () $9 is a non-empty functional set on a domain i=1 DCR”. Otherwise, non-solvable, i.e., n S° = @ for any domain D C R”. Define a topological graph G[PDES»] or G[DEPSC] in R” by V(G[PDESm]) = V(G[DEPS,]) = {S?, 1 <i < m}; E(G[PDES,,]) = E(G[DEPS‘]) = {(5?,S9) if SP( |S? #9, 1<i,7 <m} uw?) FZ: with a labeling L: 8933S), (S?,S}) € E(G[PDESm]) = E(G|DEPS¢]) > S?()S$ 4995 for 1 <i#j <™m. Similarly, if T is an invertible linear transformation on R”, then T(S®) is the solution of ith transformed equation in (PDESj,) or (DEPSC), and T($°)Q T(S?) #0 if and only if S°7) Ss? # (). Accordingly, T induces an isomorphism 7™* of graph with T* o L = LoT* holds on labeling L. We get the following result. Theorem 4.3 A system (PDESm) or (DEPS®) of partial differential equations of first order inherits an invariant GIPDESm] or G[DEPS©] under the action of invertible linear transfor- mations on R”. Such a topological graph G[PDES\,] or G[DEPSC] are said to be the G[PDES,,]-solution or G[DEPS©]-solution of systems (PDES,,) and (DEPS®), respectively. For example, the G[DEPS§]-solution of Cauchy problem Ut + Uz = 0 Ut + Uz = O : (DEPSS) uz + auz +e’ =0 ule=o = (2) is shown in Fig.8 sua sel S219 S88 Fig.8 Clearly, system (DEPS{) is contradictory because e' 4 0 for t. However, Ut tau, = 0 Ut + LUgz = 0 Up tau, +e! =0 and ult=o0 = (2) uli=o = O(@) uli=o = O(@) 20 Linfan MAO are solvable with respective solutions Sl] = {¢(x—at)}, S?] = {¢(4)} and Sl = (Oa at) — e' + 1}, and SUI SPI = {g(a — at) = 6(4)}, SPI) SP) = {4(S) = O(a — at) — e + 1}, but 17) sl = 9@. Similar to ordinary case, an interesting problem on partial differential equations is the following: Problem 4.4 For a system of (PDES,,) or (DEPS® ) of partial differential equations, deter- mine its G[PDES,]-solution or G[DEPS©]-solution. It should be noted that for an algebraically contradictory linear system Fi(21,-°- »Un,U,P1,°** :Pns) =0 Fj (a1,*-- >Un,U,P1,°** 1 Dns ) = 0, if Fi, (a1,°°- >Un,U,P1,°** Pn; ) =0 is contradictory to one of there two partial differential equations, then it must be contradictory to another. This fact enables one to classify equations in (LPDES,,) by the contradictory property and determine G[LPDES¢]. Thus if %,--- ,@ are maximal contradictory classes for equations in (LPDES), then GI[LPDESC] ~ K(@,--- ,@), ie., an I-partite complete graph. Accordingly, all G[LPDES]-solutions of linear systems (LPDES,,) are nothing else but K(@1,--- ,@;)-solutions. More behaviors on non-solvable ordinary or partial differential equations of first order, for instance the global stability can be found in references [25]-[27]. 4.3 Equation’s Combinatorics All these discussions in Sections 3 and 4.2 — 4.3 lead to a conclusion that a non-solvable system (E'S) of equations in n variables inherits an invariant G[E'S] of topological graph labeled with those of individually solutions, if it is individually solvable, i.e., equation’s combinatorics by view it with the topological graph G[ES] in R”. Thus, for holding the global behavior of a system (E'S) of equations, the right way is not just to determine it is solvable or not, but its G|ES]-solution. Such a G[ES]-solution is existent by philosophy and enables one to include non-solvable equations, no matter what they are algebraic, differential, integral or operator equations to mathematics by G-system following: Definition 4.5 A G-system (ES) of equations O;(X) = 0, 1 < i < m with constraints C is a topological graph G with labeling L : vu € V(G) > L(v) € {So,; 1 < i < m} and L: (u,v) € E(G) > L(u) (1) L(v) with L(u) (1) L(v) £0, denoted by G[ES,,], where, So, is the solution space of equation O;(X) =0 with constraints C for integers 1 <i <m. Thus, holding the true face of a thing T characterized by a system (FS) of equations needs one to determine its G-system, i.e., G[ES,,|-solution, not only solvable or not for its objective reality. Problem 4.6 Determine G[ES,,| for equation systems (ES), such as those of algebraic, Mathematics on Non-Mathematics — A Combinatorial Contribution 21 differential, integral, operator equations, or their combination, or conversely, characterize G- systems of equations for given graphs G, foe example, these G-systems of equations for complete graphs G = Ky, complete bipartite graph K(n1,n2) with ny + ng =m, path Py—1 or circuit Cm. By this view, a solvable system (E'S,,) of equations in classical mathematics is nothing else but sucha K,,-system with () L(e) #9. However, as we known, more systems of equations e€E(Km) established on characters y;, 1 <i <n for a thing T are non-solvable with contradictions if n > 2. It is nearly impossible to solve all those systems in classical mathematics. Even so, its G-systems reveals behaviors of thing T’ to human beings. §5. Geometry As what one sees with an immediately form on things, the geometry proves to be one of applicable means for portraying things by its homogeneity with distinction. Nevertheless, the non-geometry can also contributes describing things complying with the Erlangen Programme that of Klein. 5.1 Non-Spaces Let 4” = {(@1, 22,-++ ,&n)} be an n-dimensional Euclidean ( affine or projective ) space with = = a normal basis €;, 1 <i<n,@%€ 4” and let Vz, ¢V be two orientation vectors with end or initial point at Z. Such as those shown in Fig.9. => => = => zV, Vz aV Vez 4 (a) (b) Fig.9 For point VF € “”, we associate it with an invertible linear mapping js {€1,€2,-++ En} {61,€,°-+ Ea} such that p(€;) = €, 1<i<n, called its weight, i.e., (Eh, €a5°++ 1 En) = [Giglancn (1s E2s-** En)” where, [aj;],,,.,, is an invertible matrix. Such a space is a weighted space on points in #”, " denoted by (4#", 4) with w: — w(Z) = [aiy],,,,,- Clearly, if u(%1) = [ai], p(Z2) = [aii], / " tluse - Does) nxn? (4, w) = R” ( A” or P”), ie., n-dimensional Euclidean ( affine or projective space ) if and then (Z) = p(Z2) if and only if there exists a constant \ such that [a and only if [ai;] =Inxn for VF € 4". Otherwise, non-Euclidean, non-affine or non-projective nmxXn space, abbreviated to non-space. 22 Linfan MAO Notice that [aj;| = [Aol nxn ie Thus, for V%p € “”, define is an equivalent relation on invertible n x n matrixes. (Zo) = {@ € H" |W) = Au(o), A € R}, an equivalent set of points to Zp. Then there exist representatives @,,« € A constituting a partition of “” in all equivalent sets @(Z),% € #” of points, i-e., HO =| \6, with Cn (| Crz = 0 for K1,k2 € A if Ky # ko, KEA where A maybe countable or uncountable. Let u(%) = [aij] nxn equivalent set @, of points, define wa, : 0" — pa, (4) by wa, (%) = Ay. Then (4”, ra, ) is also a non-space if A, # Inxn. However, (#", 14, ) approximates to ” with homogeneity = A, for & € ,. For viewing behaviors of orientation vectors in an because each orientation vector only turns a same direction passing through a point. Thus, (4, wa,,) can be viewed as space #”, denoted by .#,",. Define a topological graph G[.4”, y] by VGA" nl) = (Gh, EN}: EGE", pl) = (60, 760) BGK, FG, AO ime © Asm A Ko} with labels L: K) €EV(GLA" pw) > 4, L: (62 ,,%62,) € EGC", ul) > 62, (02. mi Am eA. Bry? “Ung Bry Then, we get an overview on (.4”,) with Euclidean spaces ;”,« © A by combinatorics. Clearly, 4" (1) #2 = G, and Kit NA, = 0 if none of Hi, Mi. being #”". Thus, G4", pw) ~ Ky )aj-1, a star with center ”, such as those shown in Fig.10. Otherwise, G4", nu) ~ Kya), ie., |A| isolated vertices, which can be turned into Ky,4) by adding an imaginary center vertex “”. Fig.10 Mathematics on Non-Mathematics — A Combinatorial Contribution 23 Let T be an invertible linear transformation on #” determined by (2) = [aiy],,,,, (F)*- Clearly, T: 4" > 4", HR — T(4f) and T(42)NT(4E) # O if and only if KH (VAHL #0. Furthermore, one of T( #2), T(4,2) should be #”". Thus T induces an isomorphism T* from G[.4#", yu] to G[T(“"), u] of graph. Accordingly, we know the result following. Theorem 5.1 An n-dimensional non-space (#", 4) inherits an invariant Gl", p], i.e., a star Ky ja\-1 or Kya, under the action of invertible linear transformations on K”", where A is an index set such that all equivalent sets @,,,« € A constitute a partition of space H”. 5.2 Non-Manifolds Let M be an n-dimensional manifold with an alta 7 = { (Uy; ~)) | A € A}, where y, : Uy — R” is a homeomorphism with countable A. A non-manifold =M on M is such a topological space with gy: U, — R™ for integers ny, > 1, A € A, which is a special but more applicable case of non-space (R”, 4). Clearly, ifn, =n for A € A, -M is nothing else but an n-manifold. For an n-manifold M, each U) is itself an n-manifold for \ € A by definition. Generally, let M) be an n)-manifold with an alta % = { (Ur«; Prax) | K © An}, where yyx : Uxn 2 R™. A combinatorial manifold M on M issucha topological space constituted by M), » € A. Clearly, U A, is countable. If n, =n, i-e., all M) is an n-manifold for \ € A, then the union .4 of AEA My, € Ais also an n-manifold with alta A= |J A ={(U nani Pan) | KE Aa, A € AS. AEA Theorem 5.2 A combinatorial manifold M isa non-manifold on M, 1.€., M=-. Accordingly, we only discuss non-manifolds ~M. Define a topological graph G[=M] by V(GI>M}) = {Ua, € A}; E(G[AM)) = {(U,, Ung) if Uar (Urs #0, Ar, Ax € ALAd # Ad} with labels DL: UXYE V(G[-=M}]) — U), L: (Uy,,Ux,) € E(>M]) > Uy, (Urn., A FAVE A, which is an invariant dependent only on alta & of M. Particularly, if each Uy is a Euclidean spaces R*, \ € A, we get another topological graph G[R*, € A] on Euclidean spaces R*, A € A, a special non-manifold called combinatorial Euclidean space. The following result on —/ is easily obtained likewise the proof of Theorem 2.1 in [23]. 24 Linfan MAO Theorem 5.3 A non-manifold ~=M on manifold M with alta W = { (Uy; py) | X © A} inherits an topological invariant G[>M]. Furthermore, if M is locally compact, G[>M] is topological homeomorphic to G[R*, € A] if: U, = R™, AEA. It should be noted that Whitney proved that an n-manifold can be topological embedded as a closed submanifold of R?"++ with a sharply minimum dimension 2n + 1 in 1936. Applying this result, one can easily show that a non-manifold ~M can be embedded into R2"™x*1 if Nmax = max{n, € A} < oo. Furthermore, let U) itself be a subset of Euclidean space R™=+! for \ € A, then %n,,,.41 = Yr(@1, 2,°°* Ln, ) in R"™=«t!, Thus, one gets an equation Litem tl — Pr(21,%2,*°* ,Ln,) = 0 in R'™x+1_ Particularly, if A = {1,2,--- ,m} is finite, one gets a system (E'S,,) of equations Lirmaxtl — Pr(L1,£2,°** ;Ln,) = 0 Pimaetl — Pr(%1,£2,°** + Ang) = 0 (28) Devas PX (B15 L235"? * + Bayy) = O in R"™=x+1_ Generally, this system (ES,,) is non-solvable, which enables one getting Theorem 3.1 once again. 5.3 Differentiable Non-Manifolds For VM) € 7~M, if M) is differentiable determined by a system of differential equations Fi (a1, %2,°°° »Un,U, Ug, °° 5 Uan, Uryr2,°** jUsrtys***) =0 Fy2(a@1,%2,°°° >Un,U, Ug, °° * 5 Uen, Uayr2,°** Meter gt ® =0 (DESin, ) Pym, (£1, £2; > Un, U, Ux, > Urn) Uri 22> Uz en )=0 Then the system (DES‘,,) consisting of systems (DES), 1 < A < m of differential equations with prescribed initial values x;,, uo, pi, for integers 7 = 1,2,--- ,n is generally non-solvable with a geometrical figure of differentiable non-manifold —M. Notice that a main characters for points p in non-manifold =M is that the number of vari- ables for determining its position in space is not a constant. However, it can also introduces dif- ferentials on non-manifolds constrained with yx|u,qu, = Prlu,.AQ Uy for WU; Pn), (Ur, xr) € @, and smooth functions f : ~M — Rat a point p € ~M. Denoted respectively by 2p, Tp7M all smooth functions and all tangent vectorsd: 2}, — Rat a point p € =M. If y(p) € () R™) i=l a s and 8(p) = dim(() R”?)), a simple calculation shows the dimension of tangent vector space i=1 s(p) dimT,->M = 3(p) + S~ (ni — 3(p)) i=l Mathematics on Non-Mathematics — A Combinatorial Contribution 25 0 Oxi; and similarly, for cotangent vector space dimT}>M = dimT,—M with a basis with a basis ,l<i<s(p)1l<j <n; with ey = 2; ricicam| Pp { daisy l<i<s(p),1<j <n, with vy =, if 1<t<sp)}, which enables one to introduce vector field 2(-M) = (U &%j, tensor field T7(-M) = pEenaM U Tr(p,7M), where, pEenM T; (p, >M) =T,7M ®---@T,7M @T;-M @-:-@T;-M —eeeeeoeos—s—s—S 7, and connection D: 2(AM) x T7(4M) — Tz(AM) with Dxr = D(X,r) such that for VX,Y € (AM), 7,07 €T(-M), AER, f e CM(=M), (1) Dx4pytT = Dxt + fDyt and Dx(7 + Ar) = Dxt + ADxT; (2) Dx(7 @n) = Dxt@r+08Dxzt; (3) For any contraction C on T?7(4>M), Dx(C(r)) = C(Dxr). Particularly, let g € T$(-M). If g is symmetrical and positive, then =M is called a Riemannian non-manifold, denoted by (-M,g). It can be readily shown that there is a unique connection D on Riemannian non-manifold (4M, g) with equality Z(g(X,Y)) = 9(Dz,¥) + 9(X, DzY) holds. Such a D with (-~M,g), denoted by (~M,g, D) is called a Riemannian non-geometry. 0 az 0 e Now let D 9 Aue i a on (Up;y) for point p € (->M,g,D). Then ce — 0x5; a OX KI (kl) (ij) Pst) J” and pew) 1, laa, n Ag — Ag) st 2 (st)(uv) Oni OR = Daas : where g = gC) dap dai; and g(st)(uv) is an element in matrix [g*) Ga) ]-1, Similarly, a Riemannian curvature tensor R: &(AM)x B(AM) x B(AM) x &(=M) > C™%(AM) of type (0,4) is defined by R(X,Y,Z,W) = g(R(Z,W)X,Y) for VX,Y,Z,W © (AM) and with a local form R= REDE (1) (Ur) dar ® dxp, ®© dxgz ® dLyy, where ; Fae a ae POC) enh led (ae) - PEDED P(ur) (ea g(ed)(ab) ps) (st)(wr) 1 0? g(st) (49) 02 glu) (kl) A? gs) (RD) 8? g(ur)(49) =D ( OLyyOLRI OL tOXi; OLuyOXi; Ox ORI ) 26 Linfan MAO 7 r,) for Vp € 7M and g@)) = g(/—, — ), e q Fe, Bia degree of (4M, g, D) at point p € 4M (see [16] or [21] for details). which can be also used for measuring the curved Theorem 5.4 A Riemannian non-geometry (=M, g, D) inherits an invariant, 1.e., the curvature tensor R: 2(AM) x 2(AM) x # (AM) x B (4M) — C™*(AM),. 5.4 Smarandache Geometry A fundamental image of geometry ¥ is that of space consisting of point p, line L, plane P, etc. elements with inclusions P,Z 35 p and P D> L and a geometrical axiom is a premise logic function T on geometrical elements p,L,P,--- € Y with T(p, L, P,---) = 1 in classical geometry. Contrast to the classic, a Smarandache geometry SG is such a geometry with at least one axiom behaves in two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways. Thus, T(p, L, P,---) =1, ~T(p,£,P,---) =1 hold simultaneously, or 0 < =T(p,L, P,---) = ,lo,--: , Ip < 1 for an integer k > 2 in SY, which enables one to discuss Smarandache geometriy in two cases following: Case 1. T(p,L,P,---) =1A-7T(p,L,P,---) =1in SY. Denoted by U = T~1(1) c S¥, V = =T~1(1) C SY. Clearly, if U()V # 0 and there are p, L,P,---€U(\V. Then there must be T(p, L, P,---) =1 and =T(p, L, P,---) =1in UP)V, a contradiction. Thus, U()V = @ or U(\V #4 @ but some of elements p, L, P,--- € SY for T are missed in U()V. Not loss of generality, let v=Qut and v-=Ov5, k=1 i=1 where UE, Vé are respectively connected components in U and V. Define a topological graph GU, V] following: V(GU,V]) = {US;1 sk <m}|J{Vas1 <i <n}; E(GU,V)) = {(U&, Ve) if UE( VE #0, 1S k< mi <i<n} with labels L: URE V(G,V]) ~U%, VéeV(GU,V]) Ve L: (Ué,Vé) € E(GU,V]) 3 UBL) Vé, 1<k<mi1<i<n. Clearly, such a graph G[U, V] is bipartite, i.e., G[U,V] < Km with labels. Case 2. 0<-=T(p,L,P,.--)=h, bh, In <1, k > 2in SY. Denoted by Aj = hy) & SY, Ag = aT~1(Ig) Cc SY, aoe. A, = ee ot ley) Cc SY. Similarly, if A;()A; #0 and there are p, L, P,--- € A;()A;. Then there must be A;(] A; = 0 or A;(| A; 4 9 but some of elements p, L, P,--- € SY for T are missed in A; (| A, for integers 1<iAFgck. Mathematics on Non-Mathematics — A Combinatorial Contribution 27 Mi Let A; = @B Aé with AG, 1 <1 <m, connected components in A;. Define a topological 1=1 graph G[Aj;, [1, &]] following: k * V(GIAL [1 AI) = UfAei1 <i < mi}; t=1 E(GA:,[1,4]) = ( {(4G, 4%) if AG) 4B 40, 1 <1 <mi,1 <5 < mj} ie with labels tH : AZ € V(GIAs [1, A]]) + AG, AB € V(GAs, [1, &]]) > AB L: (A@, Ad) € E(G[Ai, [1,4]]) — Ag) AB, 1<i<m,1<s<m; for integers 1 <i #9 <k. Clearly, such a graph G[A;,[1, k]] is k-partite, ie., G[A;, [1,k]] < Kmy,mo,--,m, With labels. For an invertible transformation T on geometry SY, it is clear that T(p), T(L), T(P),--- also constitute the elements of SY with graphs G[U,V] and G[A;,[1,é]] invariant. Thus, we know Theorem 5.5 A Smarandache geometry SY inherits a bipartite invariant G[U, V] or k-partite G[Aj, [1, k]] under the action of its linear invertible transformations. 5.5 Geometrical Combinatorics All previous discussions on non-space (.#”, 4), non-manifold =M or differentiable non-manifold =M and Smarandache geometry S¥ allude a philosophical notion that any non-geometry can be decomposed into geometries inheriting an invariant Gl.4™, |], G[>M], G[U,V] or G[Aj, [1, Al] of topological graph labeled with those of geometries, i.e., geometrical combinatorics accordant with that notion of Klein’s. Accordingly, for extending field of geometry, one needs to determine the inherited invariants Gl#”, u], G[>M], G[U,V] or G[Aj, [1, &]] and then know geometrical behaviors on non-geometries. But this approach is passive for including non-geometry to ge- ometry. A more initiative way with realization is geometrical G-systems following: Definition 5.6 Let (%;A1), (%;A2,--: »(Gn;Am) be m geometrical systems, where Y;,, A; be respectively the geometrical space and the system of axioms for an integer 1 <i < m. A geometrical G-system is a topological graph G with labeling L : v € V(G) > Lv) € {%,G,°++ Gn} and L: (u,v) € E(G) > L(u)(\L(v) with L(u)(\L(v) 4 0, denoted by GIg, A], where ¥ = J % and A= Ai. i=1 i=1 Clearly, a geometrical G-system can be applied for holding on the global behavior of systems G,,G,-++ ,Gm. For example, a geometrical K4— {e}-system is shown in Fig.11, where, R?, 1 < i < 4 are Euclidean spaces with dimensional 3 and R}()R} maybe homeomorphic to R, R? or R? for 1 <i,j <4. 28 Linfan MAO RY RE RE RS RNR} REAR RER3 RY RIR} RS Fig.11 Problem 5.7 Characterize geometrical G-systems G[Y, A]. Particularly, characterize these ge- ometrical G'-systems, such as those of Euclidean geometry, Riemannian geometry, Lobacheushy- Bolyat-Gauss geometry for complete graphs G = Km, complete k-partite graph Km, ,mo,---,mx; path Py», or circuit Cy. Problem 5.8 Characterize geometrical G-systems G[Y, A] for topological or differentiable man- ifold, particularly, Euclidean space, projective space for complete graphs G = Km, complete k-partite graph Km, .mo,--,m,, Path Pm or circuit Cm. It should be noted that classic geometrical system are mostly Ky-systems, such as those of Euclidean geometry, projective geometry,:--, etc., also a few K2-systems. For example, the topological group and Lie group are in fact geometrical K2-systems, but neither K,,-system with m > 3, nor G # Ky m-system. §6. Applications As we known, mathematical non-systems are generally faced up human beings in scientific fields. Even through, the mathematical combinatorics contributes an approach for holding on their global behaviors. 6.1 Economics m A circulating economic system is such a overall balance input-output M(t) = LU M;(t) under- i=1 lying a topological graph G[M(t)] that there are no rubbish in each producing department. _— Whence, there is a circuit-decomposition G[M(t)] = U C's such that each output of a produc- i=1 = ing department M;(t), 1 <i <m is on a directed circuit C, for an integer 1 < s < 1, such as those shown in Fig.12. Mathematics on Non-Mathematics — A Combinatorial Contribution 29 M(t) Mit) +++ M(t) Fig.12 Assume that there are m producing departments Mj(t), Mo(t),---,Mm(t), vj output values of M;(t) for the department M;(t) and d; for the social demand. Let Fj (v1;, 2:,--+ , ns) be the producing function in M;(t). Then the input-output model of a circulating economic system can be characterized by a system of equations Generally, this system is non-solvable even if it is a linear system. Nevertheless, it is a G- system of equations. The main task is not finding its solutions, but determining whether it runs smoothly, i.e., a macro-economic behavior of system. 6.2 Epidemiology Assume that there are three kind groups in persons at time t, i.e., infected I(t), susceptible S(t) and recovered R(t) with S(t) + I(t) + R(t) = 1. Then one established the SIR model of infectious disease as follows: 7 = -RIS, — =kIS —hl, dt S(0) = So, 1(0) = Io, R(O) = 0, which are non-linear equations of first order. If the number of persons in an area is not constant, let C1,C2,---,Cm be m segregation areas with respective N,, N2,--- , Nm persons. Assume at time t, there are U;(t), V;(t) persons moving in or away C;. Thus $;(t) + L,(t) — Ui(t) + Vi(t) = N;. Denoted by c,;(t) the persons moving from C; to C; for integers 1 < i,j <m. Then S\csi(t) = Ui(t) and S° cis(t) = Vi(t). 30 Linfan MAO A combinatorial model of infectious disease is defined by a topological graph G following: V(G) = {C1, Ca, re Cm}; E(G) = {(Ci, C;)| there are traffic means from C; to Cj, 1 < 1,7 < m}, L(C,)=Nj, Lt (Cj,C;) =e, for V(Ci,0;) € E(G'), 1< 4,5 <m, such as those shown in Fig.13. Fig.13 In this case, the SIR model for areas C;, 1 <i <m turns to Lc —kI,Si, Ue Ay acid l<ism, dt S;(0) = Sio, £,(0) = Tio, R(O) = 0, which is a non-solvable system of differential equations. Even if the number of an area is constant, the SIR model works only with the assumption that a healed person acquired immunity and will never be infected again. If it does not hold, the SIR model will not immediately work, such as those of cases following: Case 1. there are m known virus 4%, %2,--- ,%m with infected rate k;, heal rate h; for integers 1 < i < m and an person infected a virus % will never infects other viruses ¥; for j Fi. Case 2. there are m varying %,%,---,%n from a virus VY with infected rate k;, heal rate h; for integers 1 < i < m such as those shown in Fig.14. QP-O---- Fig.14 However, it is easily to establish a non-solvable differential model for the spread of viruses Mathematics on Non-Mathematics — A Combinatorial Contribution 31 following by combining SIR model: S=—-k, SI S = —kSI S = —KkmSI IT=k,SI—-h I T=koSI—hoI -: T=k,SI—hmI R=h I R=hol R=hpl Consider the equilibrium points of this system enables one to get a conclusion ({27]) for globally control of infectious diseases, i.e., they decline to 0 finally if 0<5<3ch w=1 t=1 particularly, these infectious viruses are globally controlled if each of them is controlled in that area. 6.3 Gravitational Field What is the true face of gravitation? Einstein’s equivalence principle says that there are no difference for physical effects of the inertial force and the gravitation in a field small enough, i.e., considering the curvature at each point in a spacetime to be all effect of gravitation, called geometrization of gravitation, which finally resulted in Einstein’s gravitational equations ([2]) 1 RY — gfo + \gt” = —8nGT” in R*, where R4Y = RHO = Gaphot”, R= guyR'” are the respective Ricci tensor, Ricci scalar curvature, G = 6.673 x 10-8cm3/gs?, « = 8mG/c* = 2.08 x 10-*8em7! - g7! - s? and Schwarzschild spacetime with a spherically symmetric Riemannian metric 1 eL dr? — r? (dé? + sin” 0d¢”) ds? = f(t) (1 ss “2) dt? — for \ = 0. However, a most puzzled question faced up human beings is whether the dimension of the universe is really 3? if not, what is the meaning of one’s observations? Certainly, if the dimension> 4, all these observations are nothing else but a projection of the true faces on our six organs, a pseudo-truth. For a gravitational field R” with n > 4, decompose it into dimensional 3 Euclidean spaces R3, R?, ---, R’,. Then there are Einstein’s gravitational equations: 1 fet gti = anere 1 Reeve — gto R= 80TH", 1 Bie ghee er Grea for each R?, R?,--- , R?,, such as a Ky-system shown in Fig.15, 32 Linfan MAO 59 s Fig.15 where P,, Po, P3, P, are the observations. In this case, these gravitational equations can be represented by 1 RUN _ = mo R= —8nGTorN(or) with a coordinate matrix Ti11 *'* Lim *'* £13 oe T21 *'* Lam -'* 23 [zp] = Ti *'* Imm “°° Lm3 for Vp € R”, where m = aim (A R" a constant for Vp € () R™ and 2” = * for1 < i=1 i=1 m i<m,1<1i<m. Then, by the Projective Principle, i.e., a physics law in a Euclidean space any n R" ~R= U R? with n > 4 is invariant under a projection on R? from R”, one can determines t=1 its combinatorial Schwarzschild metric. For example, if m= 4, ie. t, =t,ry, =7,0, = 6 and op = ¢ for 1 < pw <™m, then ([18]) at 2Gm a IGma\ ds? = S- (1 -= +) dt? — > (1 = +) dr? — mr? (d0? + sin? 0d¢”) Cc 2 FE cr pw=1 p=1 and furthermore, ifm, = M for 1 < w~<™m, then -1 ds? = (1 — i mdt? — (1 — —) mdr* — mr*(d0? + sin? 6d¢”), cr cr which is the most enjoyed case by human beings. If so, all the behavior of universe can be realized finally by human beings. But if m < 3, there are infinite underlying connected graphs, one can only find an approximating theory for the universe, i.e., “Name named is not the eternal Name”, claimed by Lao Zi. Mathematics on Non-Mathematics — A Combinatorial Contribution 33 References 1 OU Oo ODN DH 10 11 12 13 14 15 16 17 18 19 [20 (21 [22 [23] G.Birkhoff and S.MacLane, A Survey of Modern Algebra (4th edition), Macmillan Publish- ing Co., Inc, 1977. M.Carmeli, Classical Fields—General Relativity and Gauge Theory, World Scientific, 2001. Fritz John. Partial Differential Equations(4th Edition). New York, USA: Springer-Verlag, 1982. J.L.Gross and T.W.Tucker, Topological Graph Theory, John Wiley & Sons, 1987. 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Linfan Mao, Non-solvable partial differential equations of first order with applications, submitted. F.Smarandache, Paradoxist Geometry, State Archives from Valcea, Rm. Valcea, Romania, 1969, and in Paradozist Mathematics, Collected Papers (Vol. II), Kishinev University Press, Kishinev, 5-28, 1997. F.Smarandache, Multi-space and multi-structure, in Neutrosophy. Neutrosophic Logic, Set, Probability and Statistics, American Research Press, 1998. F.Smarandache, A Unifying Field in Logics. Neutrosopy: Neturosophic Probability, Set, and Logic, American research Press, Rehoboth, 1999. W.B.V.Kandasamy, Bialgebraic Structures and Smarandache Bialgebraic Structures, Amer- ican Research Press, 2003. Wolfgang Walter, Ordinary Differential Equations, Springer-Verlag New York, Inc., 1998. Math. Combin. Book Ser. Vol.3(2014), 35-40 On Cosets and Normal Subgroups B.O.Onasanya and S.A.Ilori (Department of Mathematics, University of Ibadan, Oyo State, Nigeria) E-mail: babtu2001@yahoo.com, ilorisal@yahoo.com Abstract: The paper [5] has worked on fuzzy cosets and fuzzy normal subgroups of a group, [8] has extended the idea to fuzzy middle coset. In addition to what has been done, we make a link between fuzzy coset and fuzzy middle coset and investigate some more properties of the fuzzy middle coset. [7] made attempt with some results needing adjustment. [2], [8] and [9] have shown that if f € F(S,), the set of all fuzzy subgroups of S;, is such that Imf has the highest order and f is constant on the conjugacy classes of S,, then it is co-fuzzy symmetric subgroup of S,. Then, using some results of [5], we get another result. Key Words: Middle cosets, fuzzy normal, normal subgroups, fuzzy u-commutativity. AMS(2010): 20N25 §1. Introduction This paper seeks to contribute to the body of knowledge existing in the area of fuzzy normal subgroup without any damage to the existing one. §2. Preliminaries Definition 2.1 Let X be a non-empty set. A fuzzy subset yw of the set G is a function 1:G— [0, 1]. Definition 2.2 Let G be a group and pw a fuzzy subset of G. Then p is called a fuzzy subgroup of G if () w(ay) = min{p(x), w(y)}; (14) w(a*) = w(x); (iit) ps is called a fuzzy normal subgroup if u(xy) = (yx) for all x andy inG. Definition 2.3 Let yw be a fuzzy subset (subgroup) of X. Then, for some t in [0,1], the set fn = {x © X: p(x) > th ts called a level subset (subgroup) of the fuzzy subset (subgroup) pL. Definition 2.4 Let u be a fuzzy subgroup of a group G. Fora in G, the fuzzy left (or right) coset 1Received March 12, 2014, Accepted August 15, 2014. 36 B.O.Onasanya and S.A.Ilori ap (or a) of G determined by a and p is defined by (au)(x) = u(a~tx) (or (wa)(x) = p(xat)) for alla inG. Definition 2.5 Let ys be a fuzzy subgroup of a group G. Fora and b in G, the fuzzy middle coset aub of G is defined by (ayb)(x) = u(a~xb~*) for all x in G. Proposition 2.6 Let G be a group and yu a fuzzy subset of G. Then p is a fuzzy subgroup of G if and only if Gi, is a level subgroup of G for every t in [0, u(e)], where e is the identity of G. Theorem 2.7 Let u be a fuzzy normal subgroup of a group G. Let t € [0,1] such that t < p(e), where e is the identity of G. Then Gi is a normal subgroup of G. Remark 2.8 The paper [5] have also shown that the collection {G’,} form a chain of normal subgroups of G. Theorem 2.9 Let and X be fuzzy subgroups of G. Then they are conjugate if for someaeG we have p(a~txa) = A(x) Vr € G. Theorem 2.10 Let pp and X be any two fuzzy subgroup of any group G. Then, and X are conjugate fuzzy subgroup of G if and only if u=X. Theorem 2.11({8]) Let w be a fuzzy normal subgroup of G, Then for any g € G, u(grg~') = u(g tag) for everyx EG. §3. Some Results on Fuzzy Normal and Cosets Theorem 3.1 Let a~ya be a fuzzy middle coset of G for some a € G. Then all such a form the normalizer N() of fuzzy subgroup pu of G if and only if us is fuzzy normal. Proof The paper [5] defined the normalizer of 4 by N(u) = {a € G: p(axa') = p(x)}. Then, u(axa~!) = p(x) > p is fuzzy normal so that u(ara~!a) = p(xa) & (ax) = (xa). Conversely, let 4: be fuzzy normal and a~ja a middle coset in G. Then, for all 2 € G and some a € G, (a~*pa)(x) = w(axa~*) = u(aa~*2) = (2). This implies that pu(ara~*) = (a). Hence, {a} = N(u). Proposition 3.2 Let uu be a fuzzy normal subgroup of G by a and b. Then every fuzzy middle coset aub coincides with some left and right cosets cu and pc respectively, where c~! is the product b-ta7?. Proof By associativity in G and 2.2(iii), we have that (apib)(x) = u((a~*a)b~") = p(b~* (a~*2x)) On Cosets and Normal Subgroups 37 p(b-'a~*x) = p(c7'x) = p(axc~*) still by 2.2(iii). Thus, (ab) = cu = pc. Theorem 3.3 Let G be a group of order 2 and wu a fuzzy normal subgroup of G. Then, for some a€G and Va € G, the middle coset aya coincides with fuzzy subgroup p. Proof In the middle coset ayb, take a = b. By associativity in G, we have -1 (apia)(x) = p((a~*x)a~") . By 3.2, . Since a~! € G and G is of order 2, (ax) = p((a~")?2) = plex) = (2). Therefore, Apa = ph. Now we introduce the notion of fuzzy -commutativity. Definition 3.4 Let ys be a fuzzy subgroup of G. Two elements a and b inG are said to be fuzzy p-commutative if aub = bua. Theorem 3.5 Let be a fuzzy normal subgroup of G. Then any two elements a and b in G are fuzzy -commutative. Proof Notice that (apb)(x) = p(a~*xb~"). Then, by 2.11, u(a~ tab") = p(b-'waq") = (bua)(2). Thus, apb = bua. In [7], it is claimed that every middle coset of a group G is a fuzzy subgroup. But here is a counter example. Example 3.6 Let G = (Z4,+) and choose a= b=1 then a“! =b-1 =3. 1, ife=0=e w(x) = 0.6, otherwise. It can be seen that yu is a fuzzy subgroup of G. But the middle coset ajb defined by 1, (ap) (x) = 0.6, otherwise. ifs=2 38 B.O.Onasanya and S.A.Ilori is such that (ayb)(2) > (aub)(e). But this is a contradiction, since, usually, if is a fuzzy subgroup of a group G, p(e) > u(x) Va € G. In the following theorem, a necessary condition for middle coset of a group to be fuzzy subgroup is given. Theorem 3.7 Every middle coset ayb of a group G is a fuzzy subgroup if p is fuzzy conjugate to some fuzzy subgroup X of G. Proof Let b= a7! for some a,b € G and p and 4 be fuzzy conjugate subgroups of G. (apb)(ay~*) = (apa™*)(ay~*) = w(a~*xy*a) = Aay~*) > min{rA(a), A(y)} This implies that min{X(x), A(y)} = min{u(a~ aa), u(a~*ya)} = min{(apa~*)(x), (aa~*)(y)}- Hence, (apib)(xy~") = min{ (apd) (x), (apd) (y)}- Remark 3.8 If b = a~!, the middle coset ajia~! is a fuzzy subgroup since yp is self conjugate. Hence, the result of 3.7 generalizes the theorem 1.2.10 of [8]. Proposition 3.8 of [7] says that fuzzy middle cosets form normal subgroup of G. But here is a counter example. Example 3.9 Let G = 93 and a = (123), b = (12), x = (12), y~+ = (123). Also, define the fuzzy group pw by 1, ife =e u(x) =< 0.5, if = (123), (132) 0.3, otherwise. Then, (ayb)(xy~') = 0.3 and (ayb)(y~tx) = 1. Thus, (ayb)(ay~) A (apb)(y~*2), which implies that ayb is not fuzzy normal. We now give a characterization for aub to be fuzzy normal. It is noteworthy that [8] has shown that aya! and p are conjugates. Theorem 3.10 A fuzzy middle coset aub is fuzzy normal if and only if b= a7} and p is fuzzy normal. Proof Let ps be fuzzy normal. By definition, (apb) (ay) = w(a~*axyd~*). By 2.9 and 2.10, if we take b = a~!, ayb and pz are conjugate so that (a tayb~*) = (apd) (ay) = w(ay). On Cosets and Normal Subgroups 39 Since, 4 is fuzzy normal, way) = (yx) = w(a~*yaxb~*) = (apd) (yx). Thus, (apd) (xy) = (apd) (yz). Conversely, assume that apb is fuzzy normal. Then, (apd) (xy) = (apb)(yx). It follows from 2.10 that (apb)(zy) = p(a~*ayb~*) = play) b= a and (apb) (yx) = p(a~*yab~*) = pyr) @b=a~’. This implies that p(xy) = (apb) (xy) = (apub)(yx) = w(yr) b= a7". Hence, p is fuzzy normal and b= a"!. Proposition 3.11 Let u € F(S;,) be co-fuzzy symmetric subgroup of S,. Then pw is fuzzy normal. Proof Since p is co-fuzzy, it is constant on «IIx, the conjugate class of S$, containing I. Hence p(a~*Wxr) = p(I1) for VU € C(II) and some x € S,. By 2.2(iii), is fuzzy normal. Theorem 3.12 Every symmetric group with co-fuzzy symmetric subgroup ys is such that the level subgroups pl are normal subgroups of the symmetric group so that for t € [0,1] andt < p/(e), the collection {4} is a chain of normal subgroups of Sn. Proof Let uw € F(S;,) be such that py is co-fuzzy. Then it is fuzzy normal by 3.11. Then every level subgroup ju; (which is a subgroup of S,,) of 4 is a normal subgroup by 2.7. Then, by 2.8, the collection {4} is a chain of normal subgroups of S,. References 1] A-Rosenfeld, Fuzzy groups, J. Math. Anal. and Appl., 35(1971) 512-517. 2) Jin Bai Kim and Kyu Hyuck Choi, Fuzzy Symmetric Groups, The Journal of Fuzzy Math- ematics, Vol.3, 2(1995) 465-470. 3] L.A.Zadeh, Fuzzy sets, Inform. and Control, Vol. 8(1965), 338-353. M.Atif Misherf, Normal fuzzy subgroups and fuzzy normal series of finite groups, Fuzzy sets and system, Vol.72(1995), 512-517. 5] N.P.Mukherjee and P.Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sct., Vol.34(1984), 225-239. 40 B.O.Onasanya and S.A.Ilori P.S.Das, Fuzzy groups and level subgroups, J.Math. Anal and Appl., Vol.84(1981), 264-269. R.Nagarajan and A.Solairaju, On Pseudo Fuzzy Cosets of Fuzzy Normal Subgroups, [JCA (0975-8887), Vol.7, No.6(2010), 34-37. W.B.Vasantha Kandasamy, Smarandache Fuzzy Algebra, American Research Press, Re- hoboth 2003. W.B.Vasantha Kandasamy and D. Meiyappan, Pseudo fuzzy cosets of fuzzy subsets, fuzzy subgroups and their generalizations, Vikram Mathematical J., Vol.17(1997) 33-44. Math.Combin. Book Ser. Vol.3(2014), 41-48 On Radio Mean Number of Some Graphs R.Ponraj and S.Sathish Narayanan (Department of Mathematics, Sri Paramakalyani College, Alwarkurichi-627412, India) R.Kala (Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627012, India) E-mail: ponrajmathsQ@gmail.com, sathishrvss@gmail.com, karthipyi91@yahoo.co.in Abstract: A radio mean labeling of a connected graph G is a one to one map f from the vertex set V (G) to the set of natural numbers N such that for each distinct vertices u and v of G, d(u,v) + [Ae | >1+diam (G). The radio mean number of f, rmn(f), is the maximum number assigned to any vertex of G.The radio mean number of G, rmn (G) is the minimum value of rmn(f) taken over all radio mean labeling f of G. In this paper we find the radio mean number of some graphs which are related to complete bipartite graph and cycles. Key Words: Carona, path, complete bipartite graph, cycle, Smarandache radio mean number, radio mean number. AMS(2010): 05C78 §1. Introduction We considered finite, simple undirected and connected graphs only. Let V(G) and E(G) respectively denote the vertex set and edge set of G. Chatrand et al.[1] defined the concept of radio labeling of G in 2001. Radio labeling of graphs is applied in channel assignment problem [1]. Radio number of several graphs determined [2,7,5,9]. In this sequal Ponraj et al. [8] introduced the radio mean labeling in G. A radio mean labeling is a one to one mapping f from V (G) to N satisfying the condition Ones an 1g) (1.1) for every u,v € V (G). The span of a labeling f is the maximum integer that f maps to a vertex of Graph G. For any subgraph H < G, a Smarandache radio mean number of G on H is the lowest span taken over al such labelings of the graph G that its constraint on H is a radio mean labeling. Particularly, if H = G, such a Smarandache radio mean number is called the radio mean number of G, denoted by rmn(G). The condition (1.1) is called radio mean condition. In [8] we determined the radio mean number of some graphs like graphs with diameter three, 1Received February 19, 2014, Accepted August 20, 2014. 42 R.Ponraj, S.Sathish Narayanan and R.Kala lotus inside a circle, gear graph, Helms and Sunflower graphs. In this paper we determine radio mean number of subdivision of complete bipartite, corona complete graph with path and one point union of cycle Cg. The subdivision graph S (G) of a graph G is obtained by replacing each edge uv by a path uwv. The corona of G with H, G© HF is the graph obtained by taking one copy of G and p copies of H and joining the i*” vertex of G with an edge to every vertex in the i” copy of H. Let x be any real number. Then [x] stands for smallest integer greater than or equal to x. Terms and definitions not defined here are follow from Harary [6]. §2. Main Results Theorem 2.1 rmn(S(Kmn)) =(m4+1)(n+1)-1,m>1,n>1. Proof Let. V(S( Kaa) ) = tat Pl SS i eh A eg ed Se me ey Se} and E(S(Kmn)) = {wiwij, Wij) 2 1 <t<mj1l<j <n}. Note that diam(S(Kmjn)) = 4. Here we display S(/‘2,2) with a vertex labeling in Figure 1. Figure 1 One can easily verify that the above vertex labeling satisfies the radio mean condition. We now explain a method for labeling the vertices of S(Hm mn) where n > 3. Consider the vertex w;,;. Assign the label 2 to the vertex wm». Put the label 3 to Wm,(n—1)- Similarly for wy (n—2) we can label it by by 4. Proceeding like this w,,1 is labeled by n +1. Next we label the neighbours of um—1. Allocate the labels 2n + 3 — 7 to the vertices W(m—1),; (l<j <n). Then we move to the vertices which are adjacent to w»—2. Put the labels 3n + 4 — 7 to the vertices W(m_—2),j (1 < j <n). Proceeding like this the labels of the neighbours of u; are mn+m+1—j,1< Jj <n. Now consider the vertices u; (1 <i<m). Put the label 1 to wu. Then the vertices u; (2 <7 < m) are labeled by n+2+(n+1)(m—i). Then the integers from {mn+m+1,mn+m+2,...,mn+m-+n} are assigned to the remaining vertices in any order. Claim 1 The labeling f is a valid radio mean labeling. We must show that the condition d (t,0) + oer > 1+ diam (8 (Kmn)) =5, holds for all pairs of vertices (u,v) where u # v. Case 1. Check the pair (uj, v;). [sede sen) 504 d (uj, vj) + weet > 6. On Radio Mean Number of Some Graphs 43 Case 2. Verify the pair (ui, u;). d (uz, Uj) + Be >44+ a > 7. Case 3. Consider the pair (wi, w1,;), n > 3. E (ui) am a 14 d (ui, w1,3) + >1i+ Case 4. Examine the pair (ui, wi,;), iF 1. d (ui, wij) + Case 5. Examine the pair (uj, wi,j), we Au, n > 3. Bene > 5. d (t;, Wi,j) + > 2+2 >i] =) 2 Case 6. Check the pair (v;, w;,;). Low) Slwis)) 5 1 4 [mami ee) 5g d(v;, wij) + 5} 5 Case 7. Consider the pair (wi,;, wrt). a(n) + [Leis + 0nd] 5 94 [2#8] > Case 8. Verify the pair (v;, v;). eee So: d(uj,v;) + mn+m+1l+mn+m4+2 ee Hence rmn(S(Kmn)) = (m+1)(n+1)-1. Theorem 2.2 rmn(Kmn © P) = (m+n) (t+1),m>2,n>2,t>2. Proof Let Vika) See LS i ey eo) and Bh ee fees LX i<m1l<j <n}. Let ujus---ui be the path P! and viv}---v) be the path P*’, where 1<i<m,1<j<n. The vertex set and edge set of the corona graph Kym,» ©P, is given below. Let V (Kn © Px) = V(Kmn) U(U V(P)) U(U V(R)) and E (Kim © P) = E(Kmn) U i=1 j=l (U ECF) U (U BCRP) U {aut :1<i<gmi<j <i} U{yet:1<i<n1 <9 < th. Assign the label 1,2,--- ,m to the vertices ut,u?,--- , ui” respectively. Then we move to the path vertices of P*’. Assign the label m+ 1,m+2,---,m-+t to the vertices vt, v4,--- , uv! respectively. Then assign m+t+1,m+t+2,--+ ,m+2t to the vertices v7, v3,--- , v7? respectively. Proceeding like this until we reach the vertices of P*”. Note that vj, v¥,--- , vf received the labels m+(n—1)t+1,m+(n—1)t+2,-++,m-+nt. Again we move to the vertices of the path P?. Assign the label m+nt+1,m+nt+2,---,m+nt+t—1 to the vertices uj, uz,-+- , uz respectively. 44 R.Ponraj, S.Sathish Narayanan and R.Kala Then assign the label m+nt+t,m+nt+t+1,---,m+nt+2t—2 to the vertices u3,u3,--- , u? respectively. Proceed in the same way, assign the labels to the remaining vertices. Clearly the vertices uz”, ug’,- ++ , uz” respectively received the labels nt-+mt—t+1, nt+mt—t+2,--- ,nt+met. Finally assign the labels nt + mt+1,nt+ mt+2,...,nt+mt-+m to the vertices 21, 22,:-+ , Lm and nt+mt+m-+1,nt+mt+2,---,nt+mt+m-+n to the vertices y1, y2,.--,Yn respectively. We now check the radio mean condition for every pair of vertices. Case 1. Consider the pair (u?,u%). Subcase 1.1 7 ¥r. dluiu2) + eta ve | >6 Subcase 1.2 j =r. d(u!, ud) + ad eid. tment Se Case 2 Check the pair (u!,2,). dae ary bora ae et) oie 2 2 Case 3 Verify the pair (u?, yr). Coie J] oo. age Case 4 Examine the pair (u/, v%). d(ul v8) + [fee fur eee jae >6 Case 5 Consider the pair (v2, v’). d(uf us) + [een > 1+ fete 7 Case 6 Verify the pair (v!,x,). deo) ee) J >o+|" bt+1l+nt+mt 4 . 4? Case 7 Verify the pair (v!,y;). av se) + = sp | ee ss On Radio Mean Number of Some Graphs 45 Case 8 Consider the pair («;,2;). rene So. eee Esti d Tae] Case 9 Examine the pair (y;, y;). areal 2 > ————rr |e Aya, Yj) + Case 10 Check the pair (2;, y;). ea 2 21+ (semen ate) + | : Hence rmn(Kimn © Py) = (m+n) (t+ 1). The one point union of t cycles of length n is called the friendship graph and it is denoted by CM. Theorem 2.3 For any integer t > 2, 5t +3 if a rmn (CP) =4 5t+2 if t=3 5t+1 otherwise Proof Let uiuiuiuiuiuiui be the it” copy of the cycle CO. Identify the vertex ui (1 < i <t). It is easy to verify that ) 3 if eal diam (cf? 6 otherwise Case 1 t=2. Claim 1 rmn(C®) 4 5t +1. Suppose rmn(C?) = 5t+ 1. Let f be the radio mean labeling of cf!) for which rmn(f) = 5t +1. Then the vertices are labeled from the set {1,2,--- ,5¢+ 1}. Clearly 1 and 2 should be labeled to the vertices with a distance at least 5. The possible vertices with label 1 and 2 are indicated in Figures 2 and 3. Figure 2 46 R.Ponraj, S.Sathish Narayanan and R.Kala Figure 3 Clearly 2 and 3 are labeled at a distance at least 4 and 3 and 1 are labeled at a distance at least 5. There is no such vertex. Hence rmn(C) # St +1. Claim 2 rmn(C®) 4 5t +2. Suppose rmn(C) = 5t+ 2 then the vertices are labeled from the set {1,2,--- ,5¢+ 2}. If 1 is a label of a vertex then 3 and 4 are not labels of any vertices. Therefore the vertices are labeled from the set {2,3,--- ,5¢ +2}. Note that 2 and 3 should be labeled to the vertices which are at a distance at least 4. Therefore 2 can not be a label of the identified vertex wu‘. Suppose 2 is a label of the vertex u’,. This implies 3 should be a label of the vertex uz. Then 4 can not be a label of any of the remaining vertices. If we put the label 2 to the vertex uw, then 3 should be a label of either of the vertices u3, uz, uz. In this case also 4 can not be a label of the remaining vertices. The same fact arises when 2 is a label of the vertex uj. By symmetry, this is true for the other cases also. Hence we can not label the vertices of oe with the labels from the set {2,3,--- ,5¢+2}. Therefore rmn(C) ¢ 5t +2. Claim 3 rmn(C®) = 5t +3. The Figure 4 given below shows that the vertex labels are satisfies the radio mean condition. 9 10 13 5 ee ae), 4 11 12 8 7 Figure 4 This implies rmn(Co) = 5t+ 3. Case 2. t = 3. Claim 4 rmn(C®) > 5t +1. We observe that, for satisfying the radio mean condition, the labels 1, 2 and 3 are labels of the vertices of different cycles. Without loss of generality assume that 1 is a vertex label of the first copy of Cg, 2 is a vertex label of the second copy of Cg and 3 is a vertex label of the third copy of Cg. Note that if 1 is a label of uj or ud then 24 can not be a label. If f(u$) = 1 then 2 should be a label of u? . This implies 4 can not be a label of any of the remaining vertices. Suppose uj is labeled by 1. Then 2 is labeled by either one of the vertices u3, uz or uz. It follows that 3 should be a label of either u3, u2 or uz according as 2 is labeled. In either case 4 can not be a label of any of the vertices. Thus rmn(C®)) > 6t+1. Claim 5 rmn(C) < 5t+2. On Radio Mean Number of Some Graphs 47 The vertex labeling given in figure 5 establish that it satisfies the radio mean condition and hence rmn(C) < 5t4 2. Figure 5 Therefore rmn(C®)) = 5t+ 2. Case 3. t £ 2,3. When ¢ = 1, the vertex labels given in Figure 6 satisfies the requirements. 6 2 1 3 5 4 Figure 6 Hence rmn(Cg) = 8. Assume t > 4. Here we describe a labeling f as follows. fl) = 4 l<ist fie). = te See flag) = Mae Lest fle). = Bia TAGS fab) = 44% 1<i<t f (ui) = 5t+1. We now check whether the vertex labeling f is a valid labeling. Case 3.1 Consider the pair (uj, u‘). d(uj,u5) + [| > 2+ ay Sa) Case 3.2 Consider the pair (u4, uj). Case 3.3 Consider the pair (u4, v4). 48 R.Ponraj, S.Sathish Narayanan and R.Kala Aas ee) Sig a Se It is easy to verify that all the other pair of distinct vertices are also satisfies the radio mean condition. Hence rmn(CW) = 5t+ 1 where t F 2,3. References 1] Chartrand, Gray and Erwin, David and Zhang, Ping and Harary, Frank, Radio labeling of graphs, Bull. Inst. Combin. Appl., 33(2001), 77-85. 2] G.Chang, C.Ke, D.Kuo, D.Liu, and R.Yeh, A generalized distance two labeling of graphs, Disc. Math., 220(2000), 57-66. 3] J.A.Gallian, A Dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 19(2012) #Ds6. 4) J.R.Griggs and R.K.Yeh, Labeling graphs with a condition at distance 2, STAM J. Disc. Math., 5(1992), 586-595. 5] W.K.Hale, Frequency assignment: theory and applications, Proc. IEEE, 68(1980), 1497- 1514. 6] F.Harary, Graph Theory, Addision wesley, New Delhi (1969). 7| D.Liu and R.K.Yeh, On distance two labellings of graphs, Ars Comb., 47(1997), 13-22. 8] R.Ponraj, S.Sathish Narayanan and R.Kala, Radio mean labeling of graphs, (communi- cated). 9] J.Van den Heuvel, R.Leese, and M.Shepherd, Graph labeling and radio channel assignment, J. Graph Theory, 29(1998), 263-283. Math.Combin. Book Ser. Vol.3(2014), 49-54 Semientire Equitable Dominating Graphs B.Basavanagoud', V.R.Kulli? and Vijay V.Teli! 1. Department of Mathematics, Karnatak University, Dharwad - 580 003 2. Department of Mathematics, Gulbarga University, Gulbarga - 585 106 E-mail: b.basavanagoud@gmail.com, vijayteli22@gmail.com, vrkulli@gmail.com Abstract: The semientire equitable dominating graph SE,D(G) of a graph G = (V, E) is the graph with vertex set V US, where S is the collection of all minimal equitable dominating sets of G and with two vertices u,v € V US adjacent if u,v € D, where D is the minimal equitable dominating set or u € V(G) and v = D is a minimal equitable dominating set of G containing u. In this paper, some necessary and sufficient conditions are given for SE,D(G) to be connected and Eulerian. Finally, some bounds on domination number of SE,D(G) are obtained in terms of vertices and edges of G. Key Words: Dominating set, equitable dominating set, semientire equitable dominating graph. AMS(2010): 05C69 §1. Introduction All graphs considered here are finite, undirected with no loops and multiple edges. As usual p = |V(G)| and gq = |E(G)| denote the number of vertices and edges of a graph G = (V, E) respectively. For any graph theoretic terminology and notations we refer to Harary [3] and for more details about parameters of domination number, we refer [4] and [6]. A set D of vertices in a graph G is called a dominating set of G if every vertex in V — D is adjacent to at least one vertex in D. The domination number 7(G) of G is the minimum cardinality taken over all minimal dominating sets of G. (See Ore [7]). A subset D of V is called an equitable dominating set if for every v € V — D, there exists a vertex u € D such that uv € E(G) and |deg(u) — deg(v)| < 1. The minimum cardinality of such dominating sets is denoted by y°(G) and called the equitable domination number of G [8]. In this paper, we use this idea to introduce a new graph valued function in the field of domination theory in graphs. 1Supported by UGC-SAP DRS-II New Delhi, India 2010-2015 and the University Grants Commission, New Delhi, India. No.F.4-1/2006(BSR)/7-101/2007(BSR) dated 20th June, 2012. 2Received January 26, 2014, Accepted August 23, 2014. 50 B.Basavanagoud, V.R.Kulli and Vijay V.Teli §2. Semientire Equitable Dominating Graph Definition 1 Let G = (V,E) be a graph. Let S be the collection of all minimal equitable dominating sets of G. The semientire equitable dominating graph SE,D(G) of a graph G is the graph with vertex set V US and two vertices u,v € VUS adjacent if u,v € D, where D is a minimal equitable dominating set or u € V(G) and v = D is a minimal equitable dominating set containing wu. In Fig.1, a graph G and its semientire equitable dominating graph SE,D(G) are shown. Here D; = {1,4,5}, De = {2,4,5} and D3 = {3,4,5} are minimal equitable dominating sets of G. 1 2 LE G: 92 3 SE,D(G): 4 S527 3 Fig. 1 \ §3. Results Observation 1 In any graph G, the degree of a vertex D in SE,D(G) is the cardinality of minimal equitable dominating set D of G. The following will be useful in the proof of our results. Theorem A((2]) Let G be a graph. If D is a maximal equitable independent set of G, then D is also a minimal equitable dominating set of G. Theorem 3.1 For any nontrivial connected graph G, G C SE,D(G). Proof Let u and v be any two adjacent vertices in G but which are not adjacent in G, then we can extend the set {u, v} into maximal equitable independent set D in G which is also a minimal equitable dominating set that is u and v are adjacent vertices in SE,D(G). Hence GC SE,D(G). A subset D of V is called an equitable independent set, if for any u € D, v ¢ N(u), for all v € D— {u}. If a vertex u € V(G) be such that |deg(u) — deg(v)| > 2 for all vu € N(u) then u is in each equitable dominating set. Such vertices are called equitable isolates. Semientire Equitable Dominating Graphs 51 First we obtain a necessary and sufficient condition on a graph G such that the semientire equitable dominating graph SE,D(G) is connected. Theorem 3.2 For any nontrivial connected graph G, the semientire equitable dominating graph SE,D(G) is connected if and only if A(G) < p—1 and y°(G) > 2. Proof Let A(G) < p—1 and u, v be any two vertices in G. Then we have the following cases. Case 1 If wand v are not adjacent in G, then by Theorem 3.1, wu is adjacent to v in SE,D(G). Case 2 If u and v are adjacent in G and there is a vertex w in G which is not adjacent to both u and v, then u and v are joined by a path wwu in SE,D(G). Case 3 Let u and v are adjacent in G and w is another vertex in G which is adjacent to both u and v, then there exist two maximal equitable independent sets D, and D2 are minimal equitable dominating sets in G. Hence u and v connected through w in SE,D(G). From the above cases, we get SE,D(G) is connected. Suppose y°(G) = 1. Then every vertex of G has A(G) = p—1 and forms a minimal equitable dominating set except one vertex which is adjacent to all the other vertices in G. Therefore by definition, the semientire equitable dominating graph is disconnected, a contradiction. Hence 7°(G) 2 2. Conversely, suppose SE,D(G) is connected. On the contrary y°(G) = 1. If G is a graph having A(G) < p—1 with no equitable isolated vertices, then every vertex of G forms a minimal equitable dominating set D of G. This implies SE, D(G) is disconnected, a contradiction. Hence V(G) 2 2. Let & and k +1 be any two positive integers, 1 << k <k+1. A graph G is said to be (k,k +1) bi-regular graph, if its vertices have degree either k or k + 1. Theorem 3.3 For any unicyclic graph G without isolated vertices, then SE,D(G) is a (p+ 2,p —2) bi-regular graph. Proof Let G be a unicyclic graph of order p and contain no isolated vertices. Then from the definition of semientire equitable dominating graph, every vertex of SE,D(G) has the degree either p+ 2 or p— 2. Hence SE,D(G) is a (p + 2, p — 2) bi-regular graph. Remark 1 If T is a tree of order p, then SE,D(T) is a p-regular graph. Proposition 3.1 The semientire equitable dominating graph SE,D(G) is pK if and only if G= Ky, ; p= 2. Proof Suppose G = K,;p > 2. Then clearly each vertex of G will form a minimal equitable dominating set. Hence SE,D(G) = pK2. Conversely, suppose SE,D(G) = pK2 and G # K,. Then there exists at least one minimal equitable dominating set D containing two vertices of G. Then by the definition of semientire equitable dominating graph, D will form C3 in SE,D(G). Hence G = Kp; p > 2. 52 B.Basavanagoud, V.R.Kulli and Vijay V.Teli Theorem 3.4 Let the semientire equitable dominating graph SE,D(G) is a graph with 2p vertices and p edges if and only if G = Kp;p = 2. Proof Suppose G = K,;p > 2. Then by definition of SE,D(G), it is clear that SE,D(G) is a graph with 2p vertices and q edges. Conversely, suppose SE,D(G) is a (2p,p) graph. Then the graph pk2 is the only graph with 2p vertices and p edges. Then by Proposition 3.1, G = Kp; p > 2. Corollary 1 IfG= Ki; n> 3, then SE,D(G) = Kn+2. Theorem 3.5 If G is a connected graph with A(G) < p—1, then diam(SE,D(G)) < 2, where diam(G) is the diameter of a graph G. Proof Let G be a nontrivial connected graph and by Theorem 3.2, SE,D(G) is connected. Let u,v € V(SE,D(G)) be any two arbitrary vertices. We consider the following cases. Case 1 Suppose u,v € V(G), u and v are nonadjacent vertices in G. Then dgz,p(a)(u,v) = 1. If u and v are adjacent in G and there is no minimal equitable dominating set containing both u and v. Then there exists another vertex w in V(G), which is not adjacent to both u and v. Let D, and D2 be any two equitable dominating sets containing u,w and v,w respectively. Hence u and v are connected in SE,D(G) by a path uwwv. Thus dsp, D(C) (u,v) < 2. Case 2 Suppose u € V(G) and v ¢ V(G). Then v = D is a minimal equitable dominating set of G. Ifu € D then dgz, nia)(u,v) = 1. If u ¢ D, then there exist a vertex w € D which is adjacent to both u and v. Hence dgz, pia) (u,v) = d(u, w) + d(w, v) = 2. Case 3 Suppose u,v € V(G). Then u = D and v = D’ are two minimal equitable dominating sets of G. If D and D’ are disjoint, then every vertex in w € D is adjacent to some vertex z € D’ and vice versa. This implies that dgz,p(c)(u, v) = d(u, w) + d(w, z) + d(z,v) = 3. If D and D" have a vertex in common, then dg, p(q)(u, v) = d(u, w) + d(w, v) = 2. Thus from all these cases the result follows. The equitable dominating graph E,D(G) of a graph G = (V, E) is the graph with vertex set V UD, where D is the set of all minimal equitable dominating sets of G and with two adjacent vertices u,v € V UD if ue V and v is a minimal equitable dominating set of G containing u. Proposition 3.2({1]) The equitable dominating graph E,D(G) is pK2 if and only if G = Kp; p 2 2. Theorem 3.6 The equitable dominating graph is isomorphic to the semientire equitable domi- nating graph if and only if G is a nontrivial complete graph. Proof Let G be a nontrivial complete graph K,. Then from Proposition 3.2, EyD(G) = pK, and we have Proposition 3.1, Hence E,D(G) = SE,D(G) = pK. Semientire Equitable Dominating Graphs 53 Conversely, suppose EyD(G) = SE,D(G), Propositions 3.1 and 3.2, G must be complete graph. Hence G = Kp; p > 2. We need the following theorem for the proof of our next results. Theorem B([3]) A connected graph G is eulerian if and only if every vertex of G has even degree. Next, we prove the necessary and sufficient condition for SE,D(G) to be Eulerian. Theorem 3.7 For any graph G with no isolated vertices, SE,D(G) is Eulerian if and only if the cardinality of each minimal equitable dominating set is even. Proof Let A(G) < p—1and 7°(G) > 2, by Theorem 3.2, SE,D(G) is connected. Suppose SE,D(G) is Eulerian. On the contrary, every minimal equitable dominating set contains odd number of vertices and by observation 1, hence SE,.D(G) has a vertex of odd degree, therefore by Theorem B, SE,D(G) is not Eulerian. Hence the cardinality of minimal equitable dominating set is even. Conversely, suppose the cardinality of minimal equitable dominating set is even. Then degree of each vertex in SE,D(G) is even. Therefore by Theorem B, SE,D(G) is Eulerian. $4. Domination in SE,D(G) We first calculate the domination number of SE,D(G) of some standard class of graphs. Theorem 4.1 Let G be a graph without isolated vertices. Then, 1. if G= K,;p > 2, then 7(SE,D(K,) = p. 2. if G= Ky p;p > 1, then y(SE,D(K1,p) = 1. 3. if G=P,, p> 2, then y(SE,D(P,) = 2. 4. if G=C);p > 4, then 7(SE,D(C,) = 3. 5. if G=W,;p > 5, then 7(SE,D(W,) = 1. Theorem 4.2 Let G be any graph of order p and S = {S4,S2,53,---Sn} be the minimal equitable dominating set of G, then y(SE,D(G)) < 7(G) +|S|. Proof Let G be a connected graph. Let D = {v1, v2, v3,---u:}; 1 < i < p be the set of all minimal equitable dominating sets of G. By the definition of SE,D(G), each Si; 1<i<p is independent in SE,D(G). Hence D’ = DUS will form a dominating set in SE,D(G). Therefore 7(SE,D(G)) < |D’| =|DUS| =7(G) +|S|. Further, we get the Nordhaus-Gaddum type result for semientire equitable dominating graph. 54 B.Basavanagoud, V.R.Kulli and Vijay V.Teli Theorem 4.3 Let G be a graph such that both G and G are connected of order p > 2. Then _ 1(SE,D(G)) +(SE,D@)) < p. . 1(SE,D(G))(SE,D@)) < 2. Further, the equality holds good if and only if G = Py. References 1 B.Basavanagoud, V.R.Kulli and Vijay V.Teli, Equitable Dominating Graph (Communi- cated). K.M.Dharmalingam, Studies in Graph Theory - Equitable Domination and Bottleneck Domination, Ph.D. Thesis, Madurai Kamaraj University, Madurai, 2006. F.Harary, Graph Theory, Addison-Wesley, Reading, Mass, 1969. T.W.Haynes, $.T.Hedetniemi and P.J.Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998. T.W.Haynes, S.T.Hedetniemi and P.J.Slater, Domination in Graphs- Advanced Topics, Marcel Dekker, Inc., New York, 1998. V.R.Kulli, Theory of Domination in Graphs, Vishwa International Publications, Gulbarga, India, 2010. O.Ore, Theory of Graphs, Amer. Math. Soc. Collog. Publ., 38, Providence, 1962. V.Swaminathan and K.M.Dharmalingam, Degree equitable domination on graphs, Kragu- jevak Journal of Mathematics, Vol.35, 1(2011), 191-197. Math.Combin. Book Ser. Vol.3(2014), 55-69 Friendly Index Sets and Friendly Index Numbers of Some Graphs Pradeep G.Bhat and Devadas Nayak C (Department of Mathematics, Manipal Institute of Technology, Manipal University, Manipal-576 104, India) E-mail: pgbhat@rediff.com, devadasnayakc@yahoo.com Abstract: Let G be a graph with vertex set V(G) and edge set E(G). Consider the set A = {0,1}. A labeling f : V(G) — A, induces a partial edge labeling f* : E(G) — A, defined by f*(xy) = f(x) if and only if f(x) = f(y) for each edge xy € E(G). For i € A, let vg(i) = |{v € V(G) : f(v) = t}| and we denote epy« (i) = |{e € E(G) : f*(e) = t}]. In this paper we define friendly index number(FIN) and full friendly index number(FFIN) of graph G as the cardinality of the distinct elements of friendly index set and full friendly index set respectively and obtaining these numbers along with their sets of some families graphs. Key Words: Friendly index set, full friendly index set, friendly index number and full friendly index number, Smarandache friendly index number. AMS(2010): 05C76 §1. Introduction We begin with simple, finite, connected and undirected graph G = (V, E). Here elements of set V and E are known as vertices and edges respectively. For all other terminologies and notations we follow Harary [2]. In 1986 Cahit [1] introduced cordial graph labeling. A function f from V(G) to {0,1}, where for each edge xy, f*(xy) = |f(x) — f(y)|, ve(2) is the number of vertices v with f(v) =i and ef+(z) is the number of edges e with f*(e) = 3, is called friendly if |u¢(1) — vf(0)| <1. A friendly labeling f is called cordial if |e«(1) — ef+(0)| < 1. In [6] Lee and Ng defined the friendly index set of a graph G as FI(G) = {ler+(1) — ef+(0)|: f* runs over all friendly labeling f of G}. The concept was extended by Harris and Kwong [7] to full friendly index set for the graph G, denoted F'FI(G), defined as FFI(G) = {ey+(1) — e+ (0) : f* runs over all friendly labeling f of G}. Lee, Liu and Tan [5] considered a new labeling problem of graph theory. A vertex labeling of G is a mapping f from V(G) into the set {0,1}. For each vertex labeling f of G, a partial edge labeling f* of G is defined in the following way. 1Received February 18, 2014, Accepted August 26, 2014. 56 Pradeep G.Bhat and Devadas Nayak C For each edge uv in G, 0, if f(u) = f(v) =0 » ifftu)=fw)=1 Note that if f(u) # f(v), then the edge uv is not labeled by f*. Thus f* is a partial function from E(G) into the set {0,1}. Let v¢(0) and v¢(1) denote the number of vertices of G that are labeled by 0 and 1 under the mapping f respectively. Likewise, let e,-(0) and ey«(1) denote the number of edges of G that are labeled by 0 and 1 under the induced partial function f* respectively. In [4] Kim, Lee, and Ng defined the balance index set of a graph G as BI(G) = {le y«(1) — er«(0)| : f* runs over all friendly labelings f of G }. Definition 1.1 The corona G; © G2 of two graphs G, and G2 is defined as a graph obtained by taking one copy of G, (which has p, vertices) and p, copies of Gz and joining the i*” verter of Gy with an edge to every vertex in the i*” copy of Go. Definition 1.2 The crown C,, © ky is obtained by joining a pendant edge to each vertex of Cy. Definition 1.3 A chord of cycle Cy, is an edge joining two non-adjacent vertices of cycle Cy. Definition 1.4 The shell S;, is the graph obtained by taking n — 3 concurrent chords in cycle C,,. The vertex at which all the chords are concurrent is called the apex vertex. The shell is also called fan fy—1. Thus Sp = fn—1 = Pn-1+ A4. Definition 1.5 The wheel W,, is defined to be the join Ki +C;,. The vertex corresponding to Ky is known as apex vertex, the vertices corresponding to cycle are known as rim vertices while the edges corresponding to cycle are known as rim edges and edges joining apex and vertices of cycle are spoke edges. Definition 1.6 The helm H,, is the graph obtained from a wheel W,, by attaching a pendant edge to each rim vertex. Definition 1.7 The flower Fl, is the graph obtained from a helm H,, by joining each pendant vertex to the apex of the helm. More details of known results of graph labelings given in Gallian [3]. In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a composition. For example, 4 can be partitioned in five distinct ways 4+0,34+1,2+2,2+1+1,14+1+4+1+41. In this paper we are using the idea of integer partition of numbers. Let G be any graph with p vertices. Partition of p in to (po, p1), where po and p; are the number of vertices labeled by 0 and 1 respectively. Friendly Index Sets and Friendly Index Numbers of Some Graphs 57 §2. Main Results Here we are introducing two new parameters egy~(z) and ery«(i), which are the number of edges labeled 7 under balanced labeling and cordial labeling respectively. While proving our results, FI(G) and FFI(G) are used as below: FI(G) = {lery-(1) — ery(0)| : Ff* runs over all friendly labeling f of G}; FFI(G) = {er f+(1) — ery+(0) : F'f* runs over all friendly labeling f of G}. Theorem 2.1 Let G(V,E) be a graph with |E(G)| = ¢ and epy«(i) is the number of edges labeled 1 under the balanced labeling, where i = 0,1. Then (1) FI(G) = {lq —2(epy-(0) +epy-(1))| : the partial edge labeling Bf* runs over all friendly labeling f of G}; (2) FFI(G) = {q — 2(epy+(0) + epy-(1)) : the partial edge labeling Bf* runs over all friendly labeling f of G}. Definition 2.2 For a graph G with a subgraph H < G, the Smarandache friendly index number SFIN is the number of distinct elements runs over all labeling f : V(G) > A with friendly index set FIN(H), particularly, if H = G, such number is called friendly index number on G and denoted by FIN. Definition 2.2. The full friendly index number is the number of distinct elements in the full friendly index set and it is denoted as FFIN. We are using Theorem 2.1 to prove the following results. Theorem 2.4 In a shell graph S, with n > 4 vertices, {1,3,5,---,n—2}, ifn is odd FI(S,) = {1,3,5,---,n—1}, ifn is even Proof In a shell graph S,, |V(S,,)| =n and |E(S,,)| = 2n — 3. Case 1 n is odd. To satisfy friendly labeling, the possible compositions of n are n-1 n+l n+1n-1 : and ; . 2 2 2 2 —1 1 Consider the composition (S. “< n—-3 : ; n—5 : ; : ; ; ud +i, where i = 0,1,2,::-, ; epre(1) = 7, where j =i+1,i+2,14+3,---, 5 Therefore, ) of n. If the apex vertex labeled 0, then eg f+ (0) = lers«(1) — erp+(0)| = lon—3) —2 oS +it/)| = |n—2(i+7)|, 58 Pradeep G.Bhat and Devadas Nayak C where i = 0,1,2,---,— e n—-1 —1 and j =7+1,1+2,7+3,---, a If we consider the composition n—- a +%, where i = n— 0.1.2.--- 7 >) 9 ¥, 9 3 = (= then = 0/1;2;2+, ) of n and the apex vertex labeled 0, then egy+(0) = TED ise AO ek 8 ; if a epy+(1) = j, where if i = 0,1,2,---, : . Therefore, lene) erp. (0) = len—3)-2 (AF +i+3)| =|n-206+54+DI, n—3 n—5 where if = 0,1,2,-:- 2S then j SAO es iff = 2 then j ake, Considering all possible values of ¢ and j, we get BI(S,,) = {1,3,5,---,n—2}. Also if the apex vertex labeled 1, then FI(S;,) will be same. Case 2 n is even. To satisfy friendly labeling, the possible partition of n is (5. =) . If the apex vertex labeled 0, then, epy+(0) = > 1+i%, where i = 0,1,2,--- iS — 2; ef-(1) = j, where if i = 0, then j= 6,841,i42,-+-, 5-1) f8=1,2,--°, 5-2, then j =i+1,é+2,143,---, 5-1. Therefore, lers-(1) — ers-(0)| = | (2n — 3) 2(= 1+i+j)|=|n- (+2540), where if = 0, then j= #,i+1,é+2,---,5—1 fi=1,2,---, 5-2, then j=i+1,i+2,i+ nr eens Considering all possible values of i and 7, we get FI(S,,) = {1,3,5,---,n—1}. Also if the apex vertex labeled 1, then FI(S;,) will be same. Corollary 2.5 The graph S,, is cordial. Corollary 2.6 The friendly index set of the graph S, forms an arithmetic progression with common difference 2. Corollary 2.7 n—-1 9 * if n is odd FIN(S;,) = > ifn is even Corollary 2.8 In a shell graph S;, with n > 4 vertices, {-n+6,-n+8,—-n+10,...,.2—2}, ifn is odd FFI(S;,) = {-n+5,-n+7,-n4+9,...,n—-1}, ifn is even Friendly Index Sets and Friendly Index Numbers of Some Graphs 59 Corollary 2.9 —3, 2 1s odd FFIN(S,) = n if n is oO n—2, ifn is even Corollary 2.10 The full friendly index set of the graph S, forms an arithmetic progression with common difference 2. Example 2.11 Friendly index set of shell graph Ss is {1,3}. i, Figure 1: The shell graph Ss Table 1: Compositions of integer 5 for friendly labeling with elements of friendly index set. Compositions of integer 5 | Corresponding elements friendly index set a5 (3,2) Theorem 2.12 In a crown graph Cy © Ky with n > 3, 0,4,8,---,2n}, ifn is even FI(Cy © Ky) = : ; : {0,4,8,---,2n—2}, ifn is odd Proof Consider the crown graph C,, © ky, |V(C, © K1)| = 2n and |E(C, © K,)| = 2n. Case 1 1 is even. To satisfy friendly labeling, the possible partitions of number of vertices of cycle and pendent vertices of C,, © Ky are (n — 1,7) and (i,n — 7), where i = 0,1,2,---, = If i=0, then ery+(0) = n and ery+(1) = n. Therefore friendly index is ‘0’. If i = 1,2,3,--- es then egy (0) =n—i—1—Jj+k, where j = 0,1,2,---,i-—landk =0,1,2,--- ,i; epye(1) = 1+k, where 1 = 0,1,2,---,2-1 and k = 0,1,2,---,i such that 7+] =1i-1. Therefore, lers-(1) — ery-(0)| = [2n — 2[(n-i-1—j+k)+U+k)]] = 4G —-1-k)], 60 Pradeep G.Bhat and Devadas Nayak C where if i= 1,2,--- i then 1 =0,1,2,--,é—1 and k= 0,1,2,-+ ,i. Considering all possible values of i, | and k, we get FI={0,4,8,--- ,2n}. Case 2 n is odd. To satisfy friendly labeling, the possible partitions of number of vertices of cycle and n—- pendent vertices of C,, © Ky are (n — 1,7) and (i,n — 7), where i = 0,1,2,---, : If i=0, then ery+(0) = n and ery+(1) = n. Therefore friendly index is ‘0’. If i = —1 1,2,3,---, -——, then egy+(0) = n—i—1—Jj+k, where j = 0,1,2,--- ,i-Land k = 0,1,2,--+ ,4, epy(1) =1+k, where! =0,1,2,---,i-landk =0,1,2,--- ,isuch that 7+/ =i—1. Therefore, lery+(1) — ery-(0)| = [2n — (ni -1-F HH) + (L+H) = 4G 1-H, n—-1 where i = 1,2,.--, ,1=0,1,2,-:»,i—1and k=0,1,2,-+: yi. Considering all possible values of i, | and k, we get FI(C, © Ki) = {0,4,8,...,2n — 2}. Corollary 2.13 The graph C,, © K, is cordial. Corollary 2.14 The friendly index set of the graph Cr, © Ky forms an arithmetic progression with common difference 4. Corollary 2.15 n+l 9 2 if n is odd FIN(C,, © ky) = st 1, ifn is even Corollary 2.16 In a crown graph Cy © Ky, with n > 3, {-2n 4+ 4,-2n 4+ 8,-2n+4+ 12,...,2n}, ifn is even {-2n 4+ 6,-2n+ 10,-2n+ 14,...,2n—2}, ifn is odd FFI(C, © Ki) = Corollary 2.17 The full friendly index set of the graph C,©K, forms an arithmetic progression with common difference 4. Corollary 2.18 n, if n is even n—1, if nis odd FFIN(Cp © Ki) = Example 2.19 Friendly index set of crown graph Cs © K;, is {0,4, 8}. Theorem 2.20 In a helm graph Hy, {1,3,5,---,2n—1}, ifn is odd FI(H,) = {0,2,4,--- ,2n}, ifn is even Friendly Index Sets and Friendly Index Numbers of Some Graphs 61 y Figure 2: The crown graph Cs - ky Table 2: Compositions of integer 5 for friendly labeling with elements of friendly index set. Partition of integers 5 and 5 | Corresponding elements of friendly index set a 5) 0) and (4 8,2) and 2.9) Proof Consider the helm graph H,. |V(H,,)| = 2n+ 1 and |E(H,,)| = 3n. Case 1 n is odd. First we label the apex vertex as 0. Subcase 1.1 Ifthe compositions of rim vertices of wheel and pendent vertices of helm are (n,0) and (0,7) respectively, then er f+(0) = 2n and epy«(1) =n. Therefore ler+(1) — erg+(0)| = 2. Subcase 1.2 Ifthe compositions of rim vertices of wheel and pendent vertices of helm are (n—1, t) and (¢,n—7), where i = 1,2,3,--- ,n—1, respectively. Then egy«(0) = (n—j)+(n—i)+1, where if 7 = 1,2,3,--- i then 7 = i+ 1,¢+ 2,74 3,---,2¢ and / = 0,1,2,---,7; if SP se a, HAG St PLEO Bs MARAT = 01 Se epye(1) =k+1, where if i = 1,2,3,--- 2S, then k = 01,2, ,i—land!l=0,1,2,--- ,4; 1 Tf i ee Lis eT re eS ee ee Eh oe ee ee % 2 9 2 1=0,1,2,---,2—7. Therefore, = lerfe(1) — erg+(0)| = [2G +9 —& — 21) — nl, =4 where if i = 1,2,3,--- ==, then pp =o de RO Bit EH 01 eee and 1 an ee ere eee ee “< a en], then.) Skt oa Ss oe k = 2i —n,2i — (n — 1), 2¢ — (n—2),--- ,t-—1 and! =0,1,2,---,n—7 such that 7 +k = 21. Therefore, lery+(1) — erg+(0)| = |n + 21 — 47 +l], 62 Pradeep G.Bhat and Devadas Nayak C =i where if i = 1,2,8,-5 ==, then j = i+1,i+2,i+3,---,2i andl = 0,1,2,-+-,i; if 1 ist nS eee n=, then j =i GD gf eB And hes OTD ath ae Subcase 1.3 Ifthe compositions of rim vertices of wheel and pendent vertices of helm are (0,n) and (n,0) respectively, then er s+(0) =n and ery+(1) = 2n. Therefore lers-(1) — erg-(0)| = 2. Subcase 1.4 If the compositions of rim vertices of wheel and pendent vertices of helm are (n— (¢+1),i +1) and (t,n — 7), where i = 0,1,2,--- ,m—1 respectively. Then egy+(0) = (n—j)+(n—(¢+1)) +1, where if i = 0,1,2,--- MSS then j= 6+2,5+3,i+4,-- ,2(¢@+1) nm-1ln+1n+38 *" 2° 2 and 1 = 0,1,2,---,n — (+1); epp+(1) = K +141, where if i = 0,1,2,--- — then n-1lnt+1n+3 and 1 = 0,1,2,--- ,2; if7 = sory n—1, then 7 =14+2,14+3,14+4,---,n k=0,1,2,---,iand!=0,1,2,---,i:ifi= ano, then & = 26+1)- n, 2(¢ +1) — (n—1),2(¢+ 1) — (n — 2),--- ,¢ and 1 =0,1,2,---,n—(¢+1). Therefore lers«(1) — erp«(0)| = [8n — 2[(n — 7) + (n- (6 +1)) +k 4214 Y], =e where if i = 0,1,2,---, =, then j =i+2,i+3,i+4,---,2(i+1), k =0,1,2,---,i and St pee kik ESA So sue et es ee EO ig A ch i CR a et 2 2 k = 2(¢+1)—n,2(¢@4+ 1) —(n—1),2(¢4+1) —-(n—2),--- ,¢ andl =0,1,2,---,n—(¢+1) such that 7 + k = 2(i+1). Therefore lerp«(1) — erg+(0)| = [n+ 26-47 +41 +4], where if 7 = 0,1,2,--- — then j =i+2,1+3,1+4,---,2(@+1) andl =0,1,2,--- ,4; if n-1ln+1n+38 a aa Subcase 1.5 Ifthe compositions of rim vertices of wheel and pendent vertices of helm are vo -,n—2, then j = 14+2,1+3,i1+4,---,n andl =0,1,2,---,n—(¢+1). (0,n) and (n — 1,1) respectively, then er f-(0) =n+1 and ery+(1) = 2n — 1. Therefore Jer s«(1) — erg (0)| = 2 — 2. Considering all the above sub cases and all possible values of i, 7 and 1, we get BI(H,,) = {1,3,5,--- ,2n —1}. If we label the apex vertex as 1 and considering all possible compositions of number of vertices for friendly labeling, then also the friendly index set will be same. Case 2 1 is even. First we label the apex vertex as 0. Subcase 2.1 Ifthe compositions of rim vertices of wheel and pendent vertices of helm are (n, 0) and (0, 7) respectively, then er s+(0) = 2n and ery+(1) =n. Therefore, |ery«(1) — ery (0)| n. Friendly Index Sets and Friendly Index Numbers of Some Graphs 63 Subcase 2.2 If the compositions of rim vertices of wheel and pendent vertices of helm are (n — 7,7) and (i,n — 72), where i = 1,2,3,---,n— 1, respectively. Then egy-(0) = (n — j) + (n—i) +1, where if i = 1,2,3,--- i then j = i+1,i+2,i+3,---,2i and! = C1 a= s+15+ 2,5 +3,---,n—1, then j = i+ 1,¢+2,i+3,---,n and 1 = 0,1,2,---,n—4; epy(1) = +1, where if i = 1,2,3,--- 45, then k = 0,1,2,-.-,i—-1 ig +3,---,n—1, then k = 2s — 1,26 — (n — 1), 2% (n—2),---,4-land!=0,1,2,---,n—i such ate 2i. Therefore, |ery«(1) — ery«(0)| = |2(¢+7 —k— 21) —n|, where if 7 = 1,2,3,---, je i ee Sw | 0,1,2,---,i 1 and / = 0,1,2,3,---,2; if i = gS eg aes 5 aa --,n—1, then 7 = t+1,14+2,1+3,---,n, k = 2i—n, 2i—(n—1), 2i—(n—2),--- ,i—lLandl =0,1,2,---,n—isuch that j +k = 21. Therefore, |er+(1) — err+(0)| = |n + 21 — 47 + 4], where if ¢ = 1, 2,3,--- 5 then j =i+1,i 2,44+3,-++, 2 and 1=0,1,2,8,---,4ifF= 541,542,548, ,n—1, then 7 =i+1,1+2,14+3,---,nand/=0,1,2,---,n-17. Subcase 2.3 Ifthe compositions of rim vertices of wheel and pendent vertices of helm are (0, ) and (n, 0), respectively, then er s-(0) = n and ery+(1) = 2n. Therefore, |er s+ (0) — err (1)| =n. Subcase 2.4 If the compositions of rim vertices of wheel and pendent vertices of helm are (n — (¢+1),i+1) and (i,n—7), where i = 0,1,2,--- ,n —1, respectively. Then eg y+(0) = (n—j)+(n—(i+1)) +1, where if i = 0,1,2,--- 5 — 1, then j =i+2,i+3,i+4,---,2(¢+1) and 1=0,1,2,--- §if8= 5,541, 542,--- nl, then j =1+2,i48,--- mand! =0,1,2,---,n— (i +1); epy-(1) = K+141, where if i = 0,1,2,---,5-1, then k = 0,1,2,-:+,4; ifi = Bg tlis tle no, then k = 2(+1)—n, 2—-(n—1), 21—(n—2),--- band! = 0,1,2,---n— (t+1). Therefore, |er¢+(1) — ers+(0)| = |38n ae j)+¢(n—(¢+1)) +44 21+ LU], where if i=0,1,2,--- a then j = 1+2,1+3,1+4,--- ,2(¢+1),k =0,1,2,---,¢and/ =0,1,2,--- ,4; ifi = m5 +1,—-+2,-+-,n—1, then j =i+2,64+3,04+4,--,n, kb = 2Ui¢1)—n,2i4+ 1) —(n—1),2(¢+1) i 2),---,¢and/=0,1,2,---,n—(¢+1) such that 7 +k = 2(¢4+1). Therefore, |ery-(1) —ery+(0)| = [n+ 2i—4j +41 +4], where if i = 0,1,2,--- 5 — 1, then mae nen ee ee i oF and! =0,1,2,:-:,i;ifi=— sig thst n—2, then =14+2,14+3,1+4,---,nandl=0,1,2,---,n—(i+1). 2 Subcase 2.5 If the compositions of rim vertices of wheel and pendent vertices of helm are (0,7) and (n — 1,1), respectively, then ery-(0) = n+ 1 and er+(1) = 2n — 1. Therefore lerp(1) — eryx(0)| =n —2. Se all the above sub cases and all possible values of 7, 7 and 1, we get BI(H,,) = {0,2 ,2n}. If we label the apex ee as 1 and considering all possible compositions of number of vertices for friendly labeling , then also the friendly index set will be same. Corollary 2.21 The graph Hy, is cordial. Corollary 2.22 The friendly index set of helm graph Hy, forms an arithmetic progression with 64 Pradeep G.Bhat and Devadas Nayak C common difference 2. Corollary 2.23 In a helm graph H,,, FIN(H,,) = , n+1, Corollary 2.24 In a helm graph H,,, {-2n+5,-2n+ 7,-2n FFI(Hy) = {—2n + 6,-2n + 8, -2n ifn is odd ifn is even 9,---,2n—1}, ifn is odd 10,--- ,2n}, ifn is even Corollary 2.25 The full friendly index set of helm graph Hy, forms an arithmetic progression with common difference 2. Corollary 2.26 In a Helm graph H,, FFIN(H,) = 2n—2. Table 3: Compositions of number of rim vertices of wheel and pendent vertices of H; for friendly labeling with elements of friendly index set. Compositions of integers 5 and 5 | Corresponding elements of friendly index set 5.0) at 5 é 1,3 Example 2.27 Friendly index set of helm graph Hs is {1,3,5, 7, 9}. Theorem 2.28 In a flower graph Fly, FI(Fly) = {0,4,8,---,2n—2}, ifn is odd {0, 4, 8,--- ,2n}, ifn is even Friendly Index Sets and Friendly Index Numbers of Some Graphs 65 Figure 3: The helm graph Hs Proof Consider the flower graph Fl,. |V(Fl,)| = 2n+ 1 and |E(Fl,)| = 4n. Case 1 n is odd. First we label the apex vertex as 0. Subcase 1.1 Ifthe compositions of rim vertices of wheel and pendent vertices of helm are (n, 0) and (0, 7) respectively, then er f+(0) = 2n and er y+ (1) = 2n. Therefore, |ers«(1) — er r+ (0)| =0 Subcase 1.2 If the compositions of rim vertices of wheel and pendent vertices of helm are (n—1,1) and (i,n—1), where i = 1,2,3,--- ,n—1, respectively. Then egy+(0) = (n—j)+(n—- i) Hitl, where if i = 1,2,3,--- ASF then j= 641,642,043, ,2i and 1 =0,1,2,-+: yi: n+1n+3 n+5 DS De epye(1) =kK+1, where if i = 1,2,3,--- BS then k= 0,1,2,-- ,i—landl=0,1,2,--- ,4; n+1n+3 n+5 ifi= yo nm—1, then 7 =i+1,74+2,74+3,---,nand!l=0,1,2,--- ,n-%; ifi= 5 5 5 wrt, m—1, then k = 2i — n,2i — (n — 1), 27 — (n — 2),--- ,2-—1 and 1=0,1,2,---,n—7%. Therefore, jer s«(1) — ers-(0)| = |4n —2((n— jf) + (n—-1) +1+k+2))|, where if ¢ = 1,2,3,---,—=—, then j = i+1,i+2,443,---,2i, k = 0,1,2,---,i—1 and ate men see eee eee Ce ee ee ee ee k = 2i—n, 2i—(n—1), 2—(n—2),--- ,i—Land/l = 0,1, 2,--- ,n—isuch that j+k = 2i. Therefore lers-(1) —erp-(0)| = 4 |i —J +, whereité = 1,2,3,---,"=*, thon j = 141,442, 143,--- 2% nt+1n+3 n+5 a 9 r and!=0,1,2,--- ,wift= 1=0,1,2,---,n-i. yor n—1, then j =14+1,74+2,74+3,---,n and Subcase 1.3 Ifthe compositions of rim vertices of wheel and pendent vertices of helm are (0,7) and (n, 0) respectively, then er f+(0) = 2n and er y+ (1) = 2n. Therefore, |ers«(1) — er r+ (0)| =0. Subcase 1.4 If the compositions of rim vertices of wheel and pendent vertices of helm are (n — (t+ 1), +1) and (i,n — 7%), where « = 0,1,2,---,n — 1, respectively. Then epy-(0) = (n—j) + (n— (i +1)) +i +1, where if i = 0,1,2,---,—— dap S Dt ae ha os DG aaa and oe Oe Gah ee 2 then 7 = 7+ 2,i4+3,---,n andl =0,1,2,---,n—(i4+1); Bay ea, where if 7 = , then 7 = 66 Pradeep G.Bhat and Devadas Nayak C = 21 Ras 0,1,2,--+, 9 =>, then k =0,1,2,-+ fond! =0,1,2,--+ ,i;if¢= "54,2" Wes 1, then k = 2(¢4+1) —n,2(¢4+ 1) — (n—-1),2(¢+1)-(n icohand “01 o cums 1). Therefore |ery«(1) — ery(0)| = |4n—2((n— 7) + (n-(@4+1)) +74 21+k+1)|, where Pp 50 Roe ois i ian gma ia eg Bes Gah c= orb oa aan n-1ln+1n+3 perry 1 = 0,1,2,---,4; if = a > voeynm—, then j = 1+2,443,14+4,--- 0, k= 20¢4+1)-n,20¢+1)-(n f,2G21) (n — 2),---,¢ and 1 = 0,1,2,---,n—-(¢4+1) such that 7 +k = 2(¢+ 1). Therefore, |ery«(1) —ery+(0)| = |4@-7+1+41)|, where if i= 01,20, 2, then j = i+2,6+3,+4,---,2(i+1) and! = 0,1,2,---,i; if -~1ntl 3 i= —_ a yo n—2, then j =i+2,i43,i+4,---,nand/=0,1,2,---,n—(i+1). Subcase 1.5 If the compositions of rim vertices of wheel and pendent vertices of helm are (0,n) and (n — 1,1) respectively, then erys-+(0) = 2n and epy«(1) = 2n. Therefore, ler s+ (1) — ery-(0)| = 0. Considering all the above sub cases and all possible values of i, j and 1, we get FI(Fl,) = {0,4,8,--- ,2n—2}. If we label the apex vertex ‘1’ and consider all possible compositions of number of vertices for friendly labeling , then the balance index set will be same. Case 2. n is even. First we label the apex vertex as ‘0’. Subcase 2.1 Ifthe compositions of rim vertices of wheel and pendent vertices of helm are (n, 0) and (0, 7) respectively, then er f+(0) = 2n and er y+ (1) = 2n. Therefore, |ers«(1) — er r+ (0)| =0 Subcase 2.2 If the compositions of rim vertices of wheel and pendent vertices of helm are (n —i,i) and (i,n—i), where i = 1,2,3,---,n—1, respectively. Then egy«(0) = (n-j) + (n—- i) +i+l, where if i = 1,2,3,--- o then j =i+1,i+2,i+3,---,2¢ and 1 =0,1,2,--- yi; if P= FHL 5425430 n-1, then j=i+1i+2,8+3,---,n andl =0,1,2,--- 0-5 2 epye(1) =k +1, where if ¢ = 1,2,3,--- io then & = 0,1,2,---,i-—1 and / = 0,1,2,...,4; if n =F +L5t25 430+ .n—1, then k = 2i n,2i — (n —1),2i — (n — 2),+-»,i—1 and =0,1,2,---,n—7. Therefore, |ers«(0) — ery(1)| = |4n —2((n — 7) + (n-1) +14+k 4 21), where if i = 1,238,045, then f= 4 41d + Qe 43,-9 12k | OOo ¢— Tad t= 2S. ~ Osis gS StL Stas then], then j = it+1,i+2,i+3,..,n,k = 2i—n, 2i—(n—1), 20—(n—2),--- ,t-land/! =0,1,2,--- ,n—dsuch that 7 +k = 27. Therefore, lerp«(1) — ery«(0)| = 4-7 +1, where if 2=1, 2, 3,.. opp then j=é+1,i+2,6+3,--- 7) and 1=0,1,2,---,§f§= 541, 54+2,543,---,n—1, then j =i+1,i+2,0+3,---,nand ; 2 1=0,1,2,---,n-i. Subcase 2.3 If the compositions of rim vertices of wheel and pendent vertices of helm are (0, n) and (n, 0), respectively, then er f-(0) = 2n and ery-(1) = 2n. Therefore, |ers«(1) — err (0)| =0. Friendly Index Sets and Friendly Index Numbers of Some Graphs 67 Subcase 2.4 If the compositions of rim vertices of wheel and pendent vertices of helm are (n — (i+ 1),2+1) and (¢,n—7), where i =0,1,2,--- ,n—1, respectively. Then egy+(0) = ae eee er ee ifi =0,1,2,--- —1, then j =i+2,14+3,14+4,---,2(@¢+1) no and 1 = 0,1,2,---,7; if = me saths 5 + 2,---,n—1, then 7 = 14+ 2,14+3,---,n and 1 =0,1,2,---,n—(é+1); epy-(1) = k-+1+1, where if = 0,1,2,---,5—1, then k = 0,1,2,--- 4 and | = 0,1,2,--- ,4; fi=z.5+ +1,-4+2,--+,n—1, then & = 2(¢+1)—n,2(i+1) —(n 1),2(¢ + 1) — (n — 2),--- ,i andl = eee ,na—(i+1). Therefore, |ers-(1) — ery«(0)| = |4n — 2((n — jf) + (n— (64+1)) +i4+ 21+ +1)|, where if i = 0,1,2,---, Dae then j =i+ nivinkn abet) RaQLAn imal al aie BES pte na, then 7 =i+2,14+3,i+4,---,n,k = 2(¢+ 1) —n,2(¢4+1) — (n—1),2(@4+1) — (n—-2),--- i and | = 0,1,2,---,n—(i+1) such that 7 + k = 2(¢+1). Therefore, |ers+(1) — ery+(0)| = I4(i— j +14 1)], where if ¢ = 0,1,2,---,5-1, then j =i+2,i+3,i+4,---,2(¢+1) and i tag Pee AY, then j = i+2,i+3,i+4,---,n and PSO 5.24 17S 5 (S01 a 1), Subcase 2.5 If the compositions of rim vertices of wheel and pendent vertices of helm are (0,n) and (n — 1,1) respectively, then ery«(0) = 2n and epy«(1) = 2n. Therefore lers-(1) — ery-(0)| = 0. Considering all the above subcases and all the possible values of 7, 7 and 1, we get FI(Fl,) = {0,4,8,--- ,2n}. If we label the apex vertex ‘1’ and consider all possible compositions of number of vertices for friendly labeling, then the balance index set will be same. Corollary 2.29 The flower graph Fl, is cordial. Corollary 2.30 The friendly index set of the graph Fl, forms an arithmetic progression with common difference 4. Corollary 2.31 In a flower graph Fly, Hees ifn is odd FIN(Fln) =4 49 , ifn is even Corollary 2.32 In a flower graph Fly, {—2n + 6,-2n + 10,-2n+ 14,---,2n—2}, ifn is odd FFI(Fly) = {—2n 4+ 4,-2n + 8,-2n 4 12,--- ,2n}, ifn is even Corollary 2.33 The full friendly index set of the flower graph Fl, forms an arithmetic pro- gression with common difference 4. Corollary 2.34 In a flower graph Fly, n—-1, FFIN(Fl,) = n, ifn is even ifn is odd 68 Pradeep G.Bhat and Devadas Nayak C Example 2.35 Friendly index set of flower graph Fls is {0,4, 8}. ea Figure 4: The flower graph Fs Table 4: Compositions of number of rim vertices of wheel and number of vertices with degree two of Fl; for friendly labeling and corresponding elements of friendly index set. Compositions of integers 5 and 5 | Corresponding elements of friendly index set (5, 0) and References 1] I.Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars Combin., 23 (1987), 201-207. 2] Frank Harary, Graph Theory, Narosa Publishing House, 1989. 3] J.A.Gallian, A dynamic survey of graph labeling, The Electronics Journal of Combina- torics, 17 (2010), #DS6. 4) R.Y.Kim, S-M.Lee and H.K.Ng, On balancedness of some graph constructions, J. Combin. Math. Combin. Comp., 66 (2008), 3-16. Friendly Index Sets and Friendly Index Numbers of Some Graphs 69 [5] S-M.Lee, A.Liu and $.K.Tan, On balanced graphs, Congr. Numerantium, 87 (1992), 59-64. [6] S-M.Lee and H.K.Ng, On friendly index sets of bipartite graphs, Ars Combin., 86 (2008), 257-271. [7] W.C.Shiu, Harris Kwong, Full friendly index sets of Py x P,, Discrete Mathematics, 308(2007), 3688-3693. Math. Combin. Book Ser. Vol.3(2014), 70-88 Necessary Condition for Cubic Planar 3-Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture Mushtaq Ahmad Shah (Department of Mathematics, Vivekananda Global University (Formerly VIT Jaipur)) E-mail: shahmushtaq81@gmail.com Abstract: A conjecture of Barnette states that, every three connected cubic bipartite pla- nar graph is Hamiltonian. This problem has remained open since its formulation. This paper has a threefold purpose. The first is to provide survey of literature surrounding the conjec- ture. The second is to give the necessary condition for cubic planar three connected graph to be non-Hamiltonian and finally, we shall prove near about 50 year Barnett’s conjecture. For the proof of different results using to prove the results we illustrate most of the results by using counter examples. Key Words: Cubic graph, hamiltonian cycle, planar graph, bipartite graph, faces, sub- graphs, degree of graph. AMS(2010): 05€25 §1. Introduction It is not an easy task to prove the Barnette’s conjecture by direct method because it is very difficult process to prove or disprove it by direct method. In this paper, we use alternative method to prove the conjecture. It must be noted that if any one property of the Barnette’s graph is deleted graph is non Hamiltonian. A planar graph is an undirected graph that can be embedded into the Euclidean plane without any crossings. A planar graph is called polyhedral if and only if it is three vertex connected, that is, if there do not exists two vertices the removal of which would disconnect the rest of the graph. A graph is bipartite if its vertices can be colored with two different colors such that each edge has one end point of each color. A graph is cubic if each vertex is the end point of exactly three edges. And a graph is Hamiltonian if there exists a cycle that pass exactly once through each of its vertices. Self-loops and parallel edges are not allowed in these graphs. Barnett’s conjecture states that every cubic polyhedral graph is Hamiltonian. P.G.Tait in (1884) conjectured that every cubic polyhedral graph is Hamiltonian; this came to be known as Tait’s conjecture. It was disproved by W.T. Tutte (1946), who constructs a counter example with 46 vertices; other researchers later found even smaller counterexamples, however, none of these counterexamples is bipartite. Tutte himself 1Received November 13, 2013, Accepted August 27, 2014. Necessary Condition for Cubic Planer Three Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture 71 conjectured that every cubic 3-connected bipartite graph is Hamiltonian but this was shown to be false by discovery of a counterexample, the Horton graph [16] .David W. Barnett (1969) proposed a weakened combination of Tait’s and Tutte’s conjecture, stating that every cubic bipartite polyhedral graph is Hamiltonian this conjecture first announced in [12] and later in [3]. In [10], Tutte proved that all planar 4-connected graphs are Hamiltonian, and in [9] Thomassen extended this result by showing that every planar 4-connected graph is Hamiltonian connected, that is for any pair of vertices, there is a Hamiltonian path with those vertices as endpoints. §2. Supports for the Conjecture In [5] Holton confirmed through a combination of clever analysis and computer search that all Barnett graphs with up to and including 64 vertices are Hamiltonian. In an announcement [14,11], McKay used computer search to extend this result to 84 vertices this implies that if Barnett conjecture is indeed false than a minimal counterexample must contain at least 86 vertices, and is therefore considerable larger than the minimal counterexample to Tait and Tutte conjecture. This is not all we know about a possible counterexample; another interesting result is that of Fowler, who in an unpublished manuscript [15] provided a list of subgraphs that cannot appear in any minimal counterexample to Barnett’s conjecture. Goody in [2] consider proper subsets of the Barnett graphs and proved the following. Theorem 2.1 Every Barnett graph which has faces consisting exclusively of quadrilaterals, and hexagons is Hamiltonian, and further more in all such graphs, any edge that is common to both a quadrilateral and a hexagon is a part of some Hamiltonian cycle. Theorem 2.2 Every Barnett graph which has faces consisting of 7 quadrilaterals, 1 octagon and any number of hexagons is Hamiltonian, and any edge that is common to both a quadrilateral and an octagon is a part of some Hamiltonian cycle. In [6] Jensen and Toft reported that Barnett conjecture is equivalent to following. Theorem 2.3 Barnett conjecture is true if and only if for every Barnett graph G, it is possible to partition its vertices in to two subsets so that each induced an acyclic subgraph of G. ( This theorem is not correct) Theorem 2.4((8]) The edges of any bipartite graph G can be colored with 5(G) colors, where 6(G) is the minimum degree of vertices in G. Theorem 2.5((4]) Barnett conjecture holds if and only if any arbitrary edge in a Barnett graph is a part of some Hamiltonian cycle. Theorem 2.6({13]) Barnett conjecture holds if and only if for any arbitrary face in a Barnett graph there is a Hamiltonian cycle which passes through any two arbitrary edges on that face. Theorem 2.7((7]) Barnett conjecture holds if and only if for any arbitrary face in a Barnett graph and for any arbitrary edges e, and eg on that face there is a Hamiltonian cycle which 72 Mushtaq Ahmad Shah passes through e, and avoids e2. It is difficult to say whether any of the technique described above will aid in settling Barnett conjecture. Certainly many of them seems to be useful and worth extending. One strategy is to keep chipping away at it; if Barnett conjecture is true then Godey’s result can be extended to show that successively large and large subsets of Barnett graphs are Hamiltonian. The Grinberg’s Theorem [1] is not useful to find the counter example to Barnett’s conjecture because all faces in Barnett graphs have even number of sides. §3. New Results Supporting the Conjecture Definition 3.1 Any closed subgraph H of cubic planar three connected graph G is called complete cubic planar 3-connected subgraph H© if all possible edges in that subgraph H are drawn then it also becomes cubic planar 3-connected graph. Thus we say H© is cubic planar 3-connected graph. We illustrate by counter example following. - pl ‘ 6——__+—___0 > ° ° =) | % 4 © o o> ° o = Fig 2 Figl Fig 3 Let G be any cubic planar three connected graph as shown in Fig.1 we take its subgraph H shown in Fig.2 then we draw all possible edges in the subgraph as shown in Fig.3 the subgraph graph becomes complete cubic planar three connected H© subgraph. Definition 3.2 Any closed subgraph H of cubic planar 3-connected graph G is called complete planar n — 1 cubic 3-connected subgraph and is denoted by HC+ if all possible edges in that subgraph H are drawn then it becomes planar n — 1 cubic 3-connected graph. i.e. Only one vertex has degree two and remaining graph is cubic planar 3-connected. Illustrate by counter example. Let G be any cubic planar three connected graph as shown in Fig.4 H be its subgraph as shown in Fig.5 we draw all possible edges in the subgraph as shown in Fig.6 but still there exist a vertex having degree two only thus we say the subgraph H©*+ be its complete planar n — 1 cubic three connected graph. Necessary Condition for Cubic Planer Three Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture 73 - o ° = > + € ° >——e o ? T Lt] | 2 o os ie 4 Fig. 5 Fig. 6 Remark 3.1 A vertex can not have degree one in closed cubic planar three connected subgraphs, then it should be pendent vertex which is not possible in closed graphs so the degree of remaining vertices is two and degree cannot be more than three because it is the subgraph of cubic planar three connected graph so only possibility is that degree of remaining vertex is two. Definition 3.3 A closed subgraph H of cubic planar 3-connected graph is called complete planar n—r cubic and 8-connected if all possible edges in that subgraph H are draw then it becomes cubic planar 8-connected, but it is still planar n — r cubic and 3-connected, i,e its r vertices have degree two and remaining all vertices are cubic and three connected. It can be represented by HC'+. Lemma 3.1 A planar bipartite 3-connected and n—3 cubic is non hamiltonian. In other words a planar graph which is bipartite 3-connected and n — 3 cubic, i.e., only three of its vertices are of degree four and remaining graph is cubic then such a graph is non hamiltonian. (Only encircle vertices is of degree four and rest of the graph is cubic and 3-connected) _ a - - oo = = - « ° on _ - = = > \ - hod a. = = - —— - —s ee | ~ —e - + - +. 4 | ee = , =< FY if , i > wa - = -_ el - = od - i—as ~ = - ad -< 7” > — $ a - oe - ~ - = a — - +. - - <== « - = - - _ . - - ~~ - Fig. 7 We shall prove this result by counter example. The main aim behind the result is to prove 74 Mushtaq Ahmad Shah that if a single property is deleted in cubic planar three connected bipartite graphs then it is non-Hamiltonian. This graph can be divided in to three closed subgraphs and an isolated vertex such that these closed subgraphs are H©+ sub graphs. Later we use this result in the main theorem. Lemma 3.2 A cubic planar bipartite 2-connected graph is non-hamiltonian. It can be seen in this example. (It is not possible for me to give number of counter examples even though we can construct number of such examples) Fig.8 below is the cubic planar bipartite 2-connected graph but non-hamiltonian. * a a a 2 > 9 Oo e 9 = « e- 2 . ° . os ° ° e » ? es / | 9 ‘ | ; * \ 7 = e — oe 8 se o D>» ? s+ > Fig.8 Remark 3.2 In every cubic planar three connected bipartite graph if any one of the property is deleted then the graph is non Hamiltonian. Remark 3.3 Let [-] denotes the greatest integer function. (1) If a and b are any two positive integers then [a + b] = [a] + [}]; (2) If a is any positive integer and b is any positive real number then [a + 6] = [a] + [6]. Remark 3.4 Let G be any graph and if G is cubic planar three connected, we know that every cubic planar three connected graph, the Degree of each vertex is exactly equal to three. Thus the sum of all the degree of the Graph is 3n that is 3 d; = 3n. i=l Since each edge contributes two to the degrees thus the number of edges in the graph is d; pail in 2 2” where n is the number of vertices of the graph. Thus we conclude that if number of nodes is n 3 3 2 2 3 number of edges is 2” and if number of edges is 2" the number of nodes is £E = = x > =n. Thus we conclude that in any cubic planar three connected graph edges and nodes are connected by certain relation. Necessary Condition for Cubic Planer Three Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture 75 The number of edges of any cubic planar three connected graph is always divisible by three if we take any planar cubic three connected graph and number of edges is not divisible by three then given graph is not H© it is planar n—1 cubic and three connected, i.e., it contain a vertex of degree two only such a graph is denoted by HC+. There does not exist any two vertices of degree two because we can draw an edge between them. Lemma 3.3 The number of regions in every cubic three connected planar graph and every cubic n ; planar bipartite three connected graph of n vertices is , where n is the number of vertices 2 of the graph. Proof Since in every cubic planar three connected graph and every cubic planar three connected bipartite graph the degree of each vertices is exactly equal to three as graph is cubic. Thus the sum of all the degree of the Graph is 3n that is - d; = 3n. i=l Since each edge contributes two to the degrees thus the number of edges in these graph is n poe 2 oe where n is the number vertices of the graph. Thus we conclude that if number of vertices is n number of edges is a and if number of edges is au the number of vertices is Bi == x = =n, i.e., the number of vertices and edges are connected by certain relation. We know by Euler’s theorem on planar graphs the number of regions is equal to r=e-—v+2. Since we have a graph of n vertices as we know it is cubic planar three connected or cubic 3n planar three connected bipartite graph the number of edges in such graph’s is o as shown above, now substitute these values in equation (i) we get 3n n+A4 r=e-n+ 9 n+ 5 That proves the result. Thus from the above result we conclude that in every cubic planar three connected and every cubic planar bipartite three connected graph it is true that n+a4 2 The above result is not true for other planar graphs as we can take a counter example of three e-—v+2= connected bipartite planar graph known as Herschel graph which contain 11 vertices and 18 edges. Contain 9 regions does not satisfy the above result. Note 3.1 In every cubic planar three connected graph G and every cubic planar bipartite three connected graph Gt 76 Mushtaq Ahmad Shah 1. The order of graphs G and G* is even; n+A4 2. The number of regions in both the graphs G and Gt are equal to , where n is the total number of vertices (See lemma 3); 3. The edges and vertices in both the graphs are connected by certain relation i.e 2E K= = and V =—. 4. In G odd cycles are allowed but in G* it is bipartite thus odd cycles are not allowed. §4. Necessary Condition for a Cubic Planar 3-Connected Graph to be Non-hamiltonian Theorem A A cubic planar 3-connected graph is non-hamiltonian if the graph is divided into three closed subgraphs of any order and an arbitrary isolated vertex such that these three closed subgraphs are planar n —1 cubic three connected I.e. they are H°* subgraphs in other words a planar 3-connected graph is non-hamiltonian if these three subgraphs are such that = % 0(mod3), 3 where [-] denotes the greatest integer function and > is the number of edges in these subgraphs (Remark 3.4 above). 3 Proof Let G be any cubic planar 3-connected graph of order n number of edges is es Let us suppose that all the three closed subgraphs of G are complete closed planar cubic 3- connected, i.e. H© subgraphs then = = 0(mod3) Since odd cycles are allowed so we can take any closed subgraph of any order in such a way that these closed subgraphs are necessarily H@* first of all we shall take order of all closed subgraphs is odd if these closed subgraphs are H©+ then we have to stop the process of searching as such subgraphs exist but if such closed subgraphs are not H©+ then we try for different orders. Let order of closed subgraph be odd, i.e., n is odd say n = 2m+1 or n = 2m —1 and Since in graphs the number of vertices and edges represent positive integers so ) + [S| = 0(mod3) = 3m+1=0(mod3) => 3/3m+1 and 3/—3m => 3/38m+1-3ms 3/1, which is contradiction similarly ifn = 2m—1. We get 3/3m — 1 — 3m which gives 3/—1. This again gives contradiction. Necessary Condition for Cubic Planer Three Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture 77 Thus we conclude that : = ¥% 0(mod3) (Since such subgraphs exists we shell stop our search). Thus there exists one vertex in all the three closed subgraphs having degree 2 only (the degree cannot be one are more than three discussed above remark 4) that is these three subgraphs are H©+ subgraphs. If two vertices are of degree two we can draw an edge between them and subgraph becomes H© only that is these subgraphs are cubic planar three connected which is not possible. When graph satisfy these conditions we first of all delete all those edge which we have added in the subgraphs then we join these sub graphs together with the arbitrary isolated vertex (it must be noted that such graphs does not contain only one arbitrary vertex it may contain more than one arbitrary vertex) with those vertices of the subgraphs having degree 2 in such a way that graph becomes cubic planar three connected. since odd cycles are allowed when we start from any arbitrary vertex it is not possible to travel all the vertices once and reaches back at the stating vertex because an arbitrary vertex can be traveled only at once so we can travel at most two of these H+ subgraphs which we have joined to make the graph cubic planar three connected thus the graph so obtained is non-Hamiltonian. Now we shall illustrate the result with following graphs and prove that this condition is satisfied by these graphs. All these graphs are cubic planar three connected and non Hamiltian satisfy the above conditions Fig.9 to 26 below. - sh ae Fig.9 1) First of all let us take Tuttle graph, in which we take an encircle vertex as an isolated vertex and divide the remaining graph in three closed subgraphs not necessary of same order we shall show that these closed subgraphs are H©+ subgraphs. 78 Mushtaq Ahmad Shah Let H be its subgraph shown below H Fig. 10 Again if we draw all possible edges in this closed subgraph the subgraph becomes planar n—1 cubic and three connected, i.e., H+ subgraph as shown below and a vertex having degree two only has been shown by encircling the vertex. HE Fig. 11 Since all the three closed subgraphs of this graph are of same order so other two subgraphs have same property as discussed above. G Fig. 12 2) Now let us take younger graph of 44 vertices which is cubic planar three connected non-Hamiltonian. And isolated vertex is shown by encircle it, and remaining graph is divided into three closed subgraphs not necessarily of same order all these closed subgraphs are HC+. Let H, be its one closed subgraph as shown Necessary Condition for Cubic Planer Three Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture 79 Hi Fig.13 If we draw all possible edges in this subgraph it becomes planar n—1 cubic three connected as shown below H©+t only encircle vertex is of degree two. Ho — Fig. 14 Let another subgraph H2 of the graph is given below. H2 ‘Fig. 15 If we draw all possible edges in the subgraph it also becomes H©°+ subgraph as shown below. 80 Mushtaq Ahmad Shah Ho Fig. 16 Let another subgraph Hs of a graph is given as H3 Fig. 17 If we draw all possible edges in the subgraph it becomes H©+ subgraph as shown below Ho Fig. 18 3) Now let us take another example of cubic planar three connected non Hamiltonian graph known as Grin berg graph Of 46 vertices as shown below in which encircle vertex is an arbitrary Necessary Condition for Cubic Planer Three Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture 81 vertex. ex enaey G Fig. 19 4) Let us take its closed subgraph H; as shown below Hi Fig. 20 Now if we draw all possible edges in the subgraph it becomes H©+ subgraph as shown below Ho Fig.21 Let us take another subgraph Hy of the graph given below 82 Mushtaq Ahmad Shah Ho Fig. 22 If we draw all possible edges in the graph it becomes planar n—1 cubic and three connected as shown below Ho Fig. 23 Now again if we take another closed subgraph H3 as below H3 Fig. 24 If we again draw all possible edges in the closed subgraph it becomes H©+ subgraph as shown below Necessary Condition for Cubic Planer Three Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture 83 Ho Fig. 25 Now all other planar cubic three connected non Hamiltonian graphs satisfy this condition these graphs are shown in Fig.26 — 1 to Fig.26 — 3. Hom HT \ | 160] I IN +4 WIN = \H Barnette Basak 42 Faulkner Lederberg Graph Younger Graph Fig.26-1 84 Mushtaq Ahmad Shah —T / ® J ‘ / ~ } . f OMS ON. eeeteeeatll / ‘ \ ray, t shed hamper t? RR tara \t + yy | NIT Ty ¢-+-¢ X Y ry *s \VY Y¥/ 94 Thomassen Graph \ j \ f ‘ 4 Tutte's Graph Fig.26-3 Note 4.1 One of the most important thing regarding the cubic planar three connected non- Hamiltonian graphs which was proved by professors Linfan Mao and Yanpei Liu in 2001 [17] there exists infinite three connected non-Hamiltonian cubic maps on every surface (orient able or non-orient able) not only the above graphs but also these infinite graphs satisfy the condition which we have proved above. Now we shall show that above result is sharp. We use counter example to prove this sharpness. Below example Fig.27 is a graph which is cubic planar three connected contain Hamiltonian cycle start from Vi, V2, V3, V4,--- , Via, Vi . In this graph if we take V4 as arbitrary vertex all the three closed subgraphs are not H©+ as shown below, it is not necessary we take V4 as arbitrary vertex we can take any vertex as arbitrary vertex in such a way that remaining graph is divided into three closed subgraphs of any order but all such closed subgraphs are H+ which is not possible in this graph. G Fig. 27 Thus we conclude that all cubic planar three connected non Hamiltonian graphs can be divided in to three closed subgraphs of any order and an isolated vertex satisfying the property that Necessary Condition for Cubic Planer Three Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture 85 all the three closed subgraphs are HC+, But if graph is cubic planar three connected and Hamiltonian it is not necessary that all the three closed subgraphs satisfy H°+ property as shown below, thus the condition which we use to prove the theorem is sharp. In other words every cubic planar three connected graph which is Hamiltonian and can be divided into three closed subgraphs of any order and an isolated vertex all the three subgraphs may are may not be H©* subgraphs, but if graph is non Hamiltonian all such closed subgraphs are HC+ (There are other examples as well but it is not possible to draw all in this paper). Take a closed subgraph H and its H°*+ subgraph ; Fig.29 Fig.28 Take another closed subgraph H and its HC+ subgraph Fig.31 Fig.30 2 And finally take a closed subgraph H of order three so it is not H°+ because we cannot draw any more edge in this subgraph (these edges are parallel edges). 86 Mushtaq Ahmad Shah Fig.32 Remark 4.1 It has been discussed above that number of regions in cubic planar three connected n+A4 graphs and cubic planar three connected bipartite graphs are , thus it is necessary that every cubic planar three connected bipartite graph is non Hamiltonian if it has at least one closed subgraph which is H©* also in lemma 1. we have given a counter example of n — 3 cubic planar three connected bipartite non Hamiltonian graph satisfy H+ property. Theorem B_ Every cubic planar bipartite three connected graph is Hamiltonian (Barnett’s conjecture). Proof Since every bipartite graph is two colorable and thus without odd cycles so it contains only even cycles and number of vertices is also even, we cannot take any closed subgraph of odd order because it is not connected as odd cycles are not allowed, so every closed subgraph of cubic planar bipartite three connected graph is of even length. Thus in this type of graph n is always even, such graphs are non Hamiltonian only if there exist at least one subgraph H of any order which is planar n — 1 cubic and three connected, i.e., H°* subgraph, Then conjecture is not true because counter example can be constructed to disprove it if such a condition is satisfied. (See theorem A above) and Fig.7 of Lemma 1. Let it is true that such graphs have at least one closed subgraph of H©+ then it must satisfy the following condition 3 = ¥% 0(mod3) Since n is necessarily even i.e. order of every subgraph is even because graph is bipartite and odd cycles are not allowed. Let n = 2m. Then 3n (2m) _ 5 =3x 5 = 3m = = 0(mod3) 3m = 0(mod3) => 3/3m and3/ — 3m => 3/3m — 3m => 3/0, Necessary Condition for Cubic Planer Three Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture 87 which contradicts the given statement that = % 0(mod3). 2 Thus we conclude that there does not exist any subgraph H of cubic planar bipartite three connected graph G which is planar n — 1 cubic and three connected, i.e., which is HC+ thus there does not exists any counter example which proves that Barnett’s conjecture does not hold, thus every cubic planar bipartite three connected graph is Hamiltonian that proves the conjecture. The above theorem can be verified by Lemma 3.1 above the non Hamiltonian graph G of Lemma 3.1 can be divided into an arbitrary vertex and three closed subgraphs H°+ even though graph is bipartite( without odd cycles) three connected planar n — 3 cubic, only three vertices are of degree four and contain only even cycles. Below is the graph which is cubic planar three connected contains Hamiltonian cycle. This cannot be divided into an arbitrary vertex and three closed subgraphs H°*. ” 7 a = - >» > —<a p—- = a A bel pe “yy P P P a > 2 \ «+ , L > a» _d >—— » “ eo 7 a il be el ——# % . ¥ > . / \ y > © —_—ee i. - = f oe =< “ i i . , P« e > p— i » 3 Pe . os Sa » * oo Fig. 33 References [1] J.A.Bonday and U.S.R.Murty, Graph Theory with Applications, Macmillan, London, 1976. [2] P.R.Goodey, Hamiltonian circuit in Polytopes with even sided faces, Israel Journal of Mathematics, 22: 52-56, 1975. [3] B.Grunbaum, Polytopes, Graphs and Complezres, Bull. Amer. Math. Soc., 76:1131-1201, 1970. 88 10 11 12 13 14 15 16 [17 Mushtaq Ahmad Shah 4) A.Hertel, Hamiltonian Cycle in Sparse Graphs, Master’s thesis, university of Toronto, 2004. 5] B.D.McKay, D.A. Holton, B. Manvel, Hamiltonian cycles in cubic 3- connected bipartite planar graphs, Journal of Combinatorial Theory, series B, 38:279-297, 1985. T.R.Jensen and B.Toft, Graph Coloring Problem, J.Wiley and Sons New York, 1995. A.K.Kalmans, Konstruktsii Kubicheskih Duudolnyh 8-Svyaznyh Bez GamiltonovyhTsiklov, Sb. TrVNiii Sistem, issled. 10:64-72, 1986. D.Conig.Uber, Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre, Math- ematische Annalen, 77: 453-465, 1916. C.Thomassen, A theorem on paths in planar graphs, Journal of Graph Theory, 7:169-176, 1983. W.T.Tutte, A theorem on planar graphs, Trans. Amer. Math. Soc., 82:99-116, 1956. R.Aldred, G.Brinkmann and B.D McKay, Announcement, March 2000. D.Barnette, Conjecture 5, In W.T.Tutte ed. Recent Progress in Combinatorics, page 343 Academic press, New York, 1969. M.Braverman, Personal communication, 2004. T.Feder and C.Subi, On Barnett’s Conjecture, Published on internet http//theory.stanford. edu/tomas/bar.ps, Submitted to Journal of Graph Theory. T.Fowler, Reducible Configurations for the Barnett’s conjecture, Unpublished manuscript August 2001. Akiyama T., Nishizeki T. and Saiton N., NP-compleateness of the Hamiltonian cycle prob- lem for bipartite graphs, Journal of Information Processing, 3(2): 73-76(1980), also cited by Hertel (2005). Linfan Mao and Yanpei Liu, An approach for constructing 3-connected non-Hamiltonian cubic maps on surfaces, OR Transactions, 4(2001), 1-7. Math.Combin. Book Ser. Vol.3(2014), 89-96 Odd Sequential Labeling of Some New Families of Graphs Lekha Bijukumar (Shanker Sinh Vaghela Bapu Institute of Technology, Gandhinagar, Gujarat, India) E-mail: dbijuin@yahoo.co.in Abstract: A graph G = (V(G), E(G)) with p vertices and q edges is said to be an odd sequential graph if there is an injection f : V(G) — {0,1,--- ,q} or if G is a tree then f is an injection f : V(G) — {0,1,--- ,2qg—1} such that when each edge zy is assigned the label f(x) + f(y), the resulting edge labels are {1,3,--- ,2qg—1}. In this paper we initiate a study on some new families of odd sequential graphs generated by some graph operations on some standard graphs. Key Words: Odd sequential labeling, Smarandachely odd sequential labeling, super sub- divisions of a graph, shadow graph. AMS(2010): 05C78 §1. Introduction By a graph G = (V(G), E(G)) with p vertices and q edges we mean a simple, connected and undirected graph in this paper. A brief summary of definitions and other information is given in order to maintain compactness. The terms not defined here are used in the sense of Harary [3]. Definition 1.1 The super subdivisions of a graph G produces a new graph by replacing each edge of G by a complete bipartite graph K2,m (where m is any positive integer) in such a way that the ends of each e; are merged with two vertices of 2-vertices part of Kom after removing the edge e; from the graph G. It is denoted by SS(G). Definition 1.2 A comb is a caterpillar in which each vertex in the path is joined to exactly one pendant vertex. Definition 1.3 For a graph G, its split graph is obtained by adding to each vertex v, a new vertex v' so that v' is adjacent to every vertex that is adjacent to v inG. Definition 1.4 The shadow graph D2(G) of a connected graph G is obtained by taking two copies of G say G’ and G",, then join each vertex u' in G’ to the neighbours of the corresponding vertex u” in G”". 1Received December 9, 2013, Accepted August 31, 2014. 90 Lekha Bijukumar Definition 1.5 A bistar is the graph obtained by joining the apex vertices of two copies of star Kin by an edge. Definition 1.6([6]) Let G = (V(G), E(G)) be a graph and G1,G2,--- ,Gn be n copies of graph G. Then the graph obtained by adding an edge between G; and Gj41, fori = 1,2,---,n—1 is called a path union of G. Definition 1.7 If the vertices are assigned values subject to certain conditions then it is known as graph labeling. Graph labeling introduced by Rosa in [5] is now one of the fascinating areas of research with applications ranging from social sciences to computer science and from neural network to bio-technology to mention a few. A systematic study on various applications of graph labeling is carried out by Bloom and Golomb [1]. The famous Ringel-Kotzig [4] graceful tree conjecture and many illustrious works on it brought a tide of different labeling techniques like harmonious labeling, odd graceful labeling, edge graceful labeling etc. For detailed survey on graph labeling and related results we refer to Gallian [2]. The concept of odd sequential labeling was introduced by Singh and Varkey [7] which is defined as follows. Definition 1.8 A graph G = (V(G), E(G)) with p vertices and q edges is said to be an odd sequential graph if there is an injection f : V(G) — {0,1,---,q} or if G is a tree then f is an injection f : V(G) > {0,1,--- ,2q¢g—1} such that when each edge xy is assigned the label f(x) + fly), the resulting edge labels are {1,3,--- ,2q— 1}. The graph which admits odd sequential labeling is known as an odd sequential graph. Generally, a graph G is called Smarandachely odd sequential if there is a subset V’ Cc V(G) such that the resulting edge labels of G \ (V’) are {1,3,--- ,2q' — 1}, where q’ < q. Clearly, if V’ = 0, such a Smarandachely odd sequential graph is nothing else but an odd sequential graph. In[7] it has been also proved that the graphs such as combs, grids, stars and rooted trees of level 2 are odd sequential while odd cycles are not. Here we investigate odd sequential labeling of some new families of graphs generated by some graph operations on some standard graphs. §2. Results on Odd Sequential Labeling Theorem 2.1 The graph Cy, © nKi,m, where n is even admits odd sequential labeling. Proof Let v1,vV2,--+,Un be the vertices of C,, where n is even. Let uz be the newly added vertices in C;, to form Cy, © nKkim, where 1 <i < nand1< j < m. To define f:V(Cn ©OnKim) — {0,1,---,q@} two cases are to be considered. Case 1. n = 0(mod4) Consider the following 4 subcases: Subcase 1.1 1<i< 5 Odd Sequential Labeling of Some New Families of Graphs 91 In this caes, let f(v;) = (m+ 1) — 1) if 7 is odd and i(m + 1) — 1 if 7 is even. Subcase 1.2 sti <i<n In this case, let f(v;) = (m+ 1) -—1)+ 2 if 7 is odd and i(m +1) — 1 if 7 is even. Subcase 1.3 Lsi<Sandi<j<m In this case, let f(wij) =i(m+1) —m-+ 2(j — 1) if 2 is odd and (m+ 1)(i — 2) + 27 if 7 is even. Subcase 1.4 5tisisnandl<j<m In this case, let f(uij) = i(m+ 1) —m-+2(j — 1) if 7 is odd and (m+ 1)(¢— 2) + 2(9 +1) if 7 is even. Case 2. n = 2(mod4) Consider the following 5 subcases. | 3 Subcase 2.1 1<i< —m WN In this case, let f(v;) = (m+1)(i—1) if7 is odd, 7(m+1)—1 ifi is even and f(v;) = i(m+1)+1 n if¢=—+1. if2 a n , Subcase 2.2 yest Sn In this case, let f(v;) = (m+ 1) -—1)+2 if 7 is odd and i(m +1) — 1 if 7 is even. Subcase 2.3 l<i<S+landi<j<m In this case, let f(ui;) = i(m+1) — m+ 2(j —1) if 7 is odd, (m+ 1)(i— 2) + 2) if 7 is even and f(uij) =(m+1)(i-1)-1 fis 4+2and j=1. Subcase 2.4 i=5+2and2<j<m In this case, let f(uj;) = i(m + 1) — m+ 2(j — 1) if i is odd and (m + 1)(4 — 2) + 2(9 + 1) if 7 is even. Subcase 2.5 5t3Sisnandl<j<m In this case, let f(uij) = i(m+1) —m-+2(j — 1) if 7 is odd and (m+ 1)(@— 2) + 2(9 +1) if 7 is even. In view of the above defined labeling pattern f satisfies the conditions for odd sequential labeling. That is C;, © nK1,m is an odd sequential graph. 92 Lekha Bijukumar U8 V4 (15) U82 x i 1D - U81 69 — 28 (16) U3 U73 Te 7 A Us1 2 Q3) 03) U53 U52 U71 18 U61 U63 U62 Figure 1 Illustration 2.2 The Figure 1 shows an odd sequential labeling of Cio © 10413. Figure 2 Odd Sequential Labeling of Some New Families of Graphs 93 Theorem 2.3 The graph SS(C,,) where n is even admits odd sequential labeling. Proof Let Cy, be the cycle containing n vertices v1,v2,--- ,Un, where n is even. Let e; denotes the edge v;v;41 in Cy. For 1 <i <n each edge e; of cycle C’, is replaced by a complete bipartite graph K2,, where m is any positive integer. Let uj; be the vertices of the m vertices part of Kom where 1<i<n,1<j<m. Define f: V(SS(C,,)) — {0,1,---,q} as follows. Let f(v1) = 0 and f(vj;) = mi+ (i—2)m if2<i< %. For $+1<i<n, let f(uj) = 2mi, f(uij) =2j7 -1forl<j<m. For2<i<n,1<j<™m, let f(w;) = mit (@-2)m4 27-1. Then the above defined labeling pattern f provides odd sequential labeling for S'S(C,,) where n is even. That is for even n, S.S(C,,) admits odd sequential labeling. Illustration 2.4 The Figure 2 shows an odd sequential labeling of SS(Cg) with Ko,3. Theorem 2.5 The split graph of comb is an odd sequential graph. Proof Let {vj,1 <i <n} and {vj,1 <i < n} be the vertices of comb in which {vj,1 < i <n} are the pendant vertices. Let {uj,1 <7 <n} and {ui,1 <i <n} be the newly added vertices and let G be the split graph of comb. Define f : V(G) — {0,1,--- ,q} as follows. Let f(v;) = 6i — 4 if 2 is odd and 6% — 3 if 7 is even, and f(v{) = 1, f(vs) = 15. Let f(vuj) = 6i — 3 if i is odd, 7 4 1,3, and 62 — 4 if 7 is even. Let f(u;) = 6% — 6 if 7 is odd, and 6i — 7 if 7 is even, and f(u,) = 3, f(ug) =11. Let f(ul) = 6i —7 if 7 is odd, i £.1,3 and 61 —6 if 2 is even. Then the above defined function provides an odd sequential labeling for the split graph of comb. That is, split graph of comb is an odd sequential graph. Illustration 2.6 The Figure 3 shows an odd sequential labeling of split graph of a comb. U1 U2 U3 UA U5 Figure 3 94 Lekha Bijukumar Theorem 2.7 D2(Comb) admits sequential labeling. Proof Consider two copies of comb G and G2. Let {uj,1 <i <n} and {vj,1 <i <n} be the vertices of comb Gy and {u;,1 <7 < mn} and {uj,1 <i <n} be the vertices of Go. Let G be the shadow graph of the comb. Define f : V(G) — {0,1,...,q} as follows. For 1 <i <n, let f(u;) = 8i — 8 if i is odd and 8 — 7 if 7 is even. For 1 <i < n, let f(v{) = 81 —7 if 7 is odd, and 81 —8 if 7 is even. For 1 <i <n, let f(u;) = 8¢—4 if i is odd and 8i —5 ifi is even. For 1 <i <n, let f(ui) = 8 —5 if2 is odd and 8 — 4 if 7 is even. In view of the above defined labeling pattern f satisfies the conditions of odd sequential labeling. That is the D2(Comb) admits odd sequential labeling. Illustration 2.8 The following Figure 4 shows an odd sequential labeling of D2(Comb). / y Vy V2 U3 U4 Us V6 Figure 4 Theorem 2.9 The graph D2(Bn») is an odd sequential graph. Proof Consider two copies of B(n,n) say Bi(n,n) and Bo(n,n). Let {v1, v2, v1;, ¥2;,1 < j <n} and {u, ue, u1j,u2j,1 <7 <n} be the vertices of Bi(n,n) and Bo(n,n) where v1, v2 and uy, U2 are the respective apex vertices. Let D2(Bn,,) be the shadow graph of By(n,n) and Bo(n,n). Define f : V(D2(Bnn)) > {0,1,...,¢q} as follows. Let f(vi) = 0, f(v2) = 8n+1, f(ur) = 4n, f(ue) = 8n4+ 3, f(z) = 4 -—1)4+1 if legen, flv) =47 if l <j cn-1, (vn) =4(n4+1), f(um;) = 479 -lifl <j <n and f(ua;) =4(n+g4+1)ifl<j<n. The above defined function f provides an odd sequential labeling for D2(By,,). That is D2(Bnin) is an odd sequential graph. Illustration 2.10 The following Figure 5 shows an odd sequential labeling of D2(B4,4). Odd Sequential Labeling of Some New Families of Graphs 95 U11 U12 U13 U14 Figure 5 Theorem 2.11 Path union graph of even cycle Cy, is an odd sequential graph. Proof Consider k copies of even cycle C,,. Let v;,; be the vertices of C, where 1 <i<k and 1 <j <n. Without loss of generality let each copy of C, is joined to its succeeding one by the edge v;,10;41,1; 1 <i <k—1. Let G be the path union graph of k copies of even cycle Cy. Define f : V(G) — {0,1,--- ,q} as follows. Case 1. k is odd. In this case, let f(via) = (n+ 1)(@—-1), f(viz) = (n+1)\G@-1)4+1. Forl<i<k 2 4 adie gS —, let f(vig) = in +1)—j4+2. Forl<i< k and "= Sree ee f(vi7) =i(n+1) —7 if 7 is odd, and i(n+ 1) — 7 + 2 if 7 is even. Case 2. k is even. n—2 si 2 and <j<n, let f(uj;) =i(n+1)— 7 —2 if 7 is odd, and i(n + 1) — j if 7 is even. In this case, for 1 <i<kand1l<j< let f(uij) =i(n+1)-j-2, forl<i<k The above described function satisfies all the conditions of odd sequential labeling. That is, path union graph of even cycle C,, is an odd sequential graph. Illustration 2.12 The following Figure 6 shows an odd sequential labeling of 4 copies of cycle Cg. 96 Lekha Bijukumar V1,7 V1.6 VLA oe Figure 6 §3. Concluding Remarks This paper presents 6 families of odd sequential graphs which are generated by some graph operations on some standard graphs. To investigate similar results for other graph families in the context of different labeling techniques is an open area of research. References 1 (6 G.S.Bloom and $.W.Golomb, Applications of numbered undirected graphs, Proceeding of IEEE,65(4) (1977) 562-570. J.A.Gallian, A dynamic survey of graph labeling, The Electronics Journal of Combina- torics, 18(2011) 1DS6. F.Harary, Graph theory, Addison Wesley, Reading, Massachusetts, 1972. 4) G.Ringel, Problem 25, Theory of graphs and its applications, Proc.Symposium Smolenice 1963, Prague (1964),162. A.Rosa, On certain valuation of the vertices of a graph, Theory of graphs, International Sym- posium, Rome, July (1966), Gorden and Breach, New York and Dunod Paris(1967),349- 355. S.C.Shee and Y.S.Ho, The cordiality of the path union of n copies of a graph, Discrete Mathematics, 151,(1996),221-229. [7] G.S.Singh and T.K.M.Varkey, On odd sequential and bi sequential graphs, Preprint. Math.Combin. Book Ser. Vol.3(2014), 97-108 Mean Labelings on Product Graphs Teena Liza John and Mathew Varkey T.K (Department of Mathematics, T.K.M College of Engineering, Kollam-5, Kerala, India) E-mail: teenavinu@gmail.com, mathewvarkeytk@gmail.com Abstract: Let G be a (p,q) graph and let f : V(G) — {0,1,--- ,q} be an injection. Then G is said to have a mean labeling if for each edge wv, there exists an induced injective map f* : E(G) > {1,2,--- ,q} defined by f'(uv) = fu) + #@) if f(u) + f(v) is even, and _ fr iort if f(u) + f(v) is odd We extend this notion to Smarandachely near m-mean labeling if for each edge e = uv and an integer m > 2, the induced Smarandachely m-labeling f* is defined by +.) — | SM +f) pl) = |) A graph that admits a Smarandachely near mean m-labeling is called Smarandachely near m-mean graph. The graph G is said to be a near mean graph if the injective map f : V(G) > {1,2,---,q—1,q+1} induces f* : E(G) — {1,2,--- ,q} which is also injective, defined as above. In this paper we investigate the direct product of paths for their meanness and the Cartesian product of P, and K, for its near-meanness. Key Words: Smarandachely near m-labeling, Smarandachely near m-mean graph, mean graph, near-mean graph, direct product, Cartesian product. AMS(2010): 05C78 §1. Introduction By a graph we mean a finite, undirected graph without loops or multiple edges. For all the ter- minology and notations in graph theory we follow [2] and [5] and for the terminology regarding labeling we follow [1]. The vertex set and edge set of a graph G are denoted by V(G) and E(G) respectively. The direct product of G and H is denoted by G x H and is defined as a graph with vertex set V(G) x V(H) and edge set {(g,h), (9',h')/gg' € E(G) and hh’ € E(H)}. The Cartesian product of G and A is denoted by GOA and is defined as a graph with 1Received October 1, 2013, Accepted September 2, 2014. 98 Teena Liza John and Mathew Varkey T.K vertex set V(G) x V(H) and edge set {(g,h), (g’, h’)/either (g = g’ and h adj h’) or (g adj g’ and h = h’)}. The concept of mean labeling was introduced in [6] and the notion of near-mean labeling was introduced in [3]. In [4], various product graphs are proved as near-mean graphs. §2. Direct Product of Graphs Definition 2.1 The direct product of G and H is the graph denoted by G x H, whose vertex set is V(G) x V(A) and for which vertices (g,h) and (g',h’) are adjacent precisely if gg’ © E(G) and hh’ € E(H). Thus V(GxH) = {(g,h)/g € V(G) andhe€ V(H)} {(g,h)(9',h')/gg' € E(G) and hh’ € E(H)} Bs op) x = | Remark 2.1 P,, x P, is a disconnected graph with two components. Direct product is both commutative and associative. The maps (21,22) + (2,41) and ((%1,%2),%3) (#1(x2, %3)) give rise to the following isomorphisms Gy x Go & Go x Gi, (Gy x G2) x G3 = Gy x (Go x G3) Theorem 2.1 P3 x Py, is a mean graph when m > 3 and is odd. Proof Let uj; i = 1,2,3;7 = 1,2,--- ,m be the vertices of P3 x P,. Note that this graph has 3m vertices and 4(m — 1) edges. Define f : V(P3 x Pn) — {0,1,---,q} such that flu) = 0 23-3 7 =3,5,--,m f(us;) = e j=2 f(uj2)+5—-k 37 =4,6,...,.m—1Lk =1,2,3;1,2,3;1,2,3--- 2(j — 1) 7 =2,4,--»,m—1 fluz;) = i 1) j=l f(u2,j-2) + 57 =3,5,-++,m ay 7 =1,3,---,m f (us;) = lm j=2 f(us3,j-2) +4 3579 =4,6---,m-1 It can be easily verified that f is one one which induces the edge labels f*(E(P3 x Pm)). Hence the theorem. Example 2.1 The Fig.1 following shows the mean labeling of P3 x Py. Mean Labelings on Product Graphs 99 P3 Fig 1 Theorem 2.2 Ps x Py, admits mean labeling when m > 7 and is odd. Proof Let uj, i= 1,2,---,5 and j = 1,2,---,m be the vertices of Ps x Pm. Consider f : V(Ps x Pn) — {0,1,--- ,q} which is defined as f(un) = 0 fis) 2 G20. F285 fg) = f@ijga) +8) 2H FH 8 bes f(u2) = i, 1=2,4 F(uix) = f(Uin-2)+8, t= 2,4;k =4,6,---,m—1 And when i = 1,3,5, f(ui2) = f(usm) + %; when i = 2,4, f(ui) = f(uam-—1) +i — 1; when i= 1,2,--- ,5;51=3,4,---,m, f(uu) = f(uii—2) +8. From the definition of labelings on V(Ps x P,,), we can infer that the vertex labels are in an increasing sequence. That is the sequence such as: For Jj = 1,3, nee ym”, (u1;); (u3j) and (u5;)3 for Jj = 2,4, Pe TU A, (u2;); (ua;) and for k = 2,4,---,m-—1, (uik), (Usk), (Usp); for k = 1,3,---,m, (uaz) and (ua,), Occur as an arithmetic progression. Also we have f(uu) = 0, f(usi)=1 f(usi) = 3, f(u22) =2 f(ua2) = 4 Hence f, is one- one with f> = {1,2,--+,q}. Remark 2.2 P,, x P,, are not mean graphs for all m. Since P, x P,, being a disjoint union of two P,,, paths, it has 2(m — 1) edges on 2m vertices. This implies that the number of edges is less than the number of vertices by 2. Hence we cannot label them with {0,1,--- ,q}. 100 Teena Liza John and Mathew Varkey T.K Conjecture 2.1 Form even P3 x Py and Ps x Pm are not mean graphs. §3. Cartesian Product of Graphs Definition 3.1 Let G and H be graphs with V(G) = V, and V(H) = V2. The cartesian product of G and H is the graph GOH whose vertex set is Vi x V2 such that two vertices u = (x,y) and v = (2’,y’) are adjacent if and only if either x = x' and y is adjacent to y’ in H ory=y' and x is adjacent to x' in G. That is u adj v in GOH whenever [x = x’ and y adj y'| or [y = y’ and x adj x}. Definition 3.2 Let P, be a path on n vertices and K4 be the complete graph on 4 vertices. The cartesian product of P, and K4 is P,OK4, with 4n vertices and 10n — 4 edges. Theorem 3.1 P,OK4 is a near mean graph. Proof Let G= P,OK4 with V(G) = {war, uia, wiz, Wia/t = 1,2,--- ,n}. Define f : V(G) > {0,1,---,q¢—1,¢+1} such that f(un) = 0, f (war) = 5(2i — 1), pH 2A = 5(2i—2), i#1, odd f(ui2) = 10(i-1)+2 fluis) = 5(2’—1) + (-1)'2 5(2i—1) +3, i odd f(uia) = 5(2i — 3) +4 7 even The edge labels induced by f are as follows: When 7 is even, f*(uaui2) = te + f(ui2) +1 1 = j)sa1—0 +5012) +244 = 10:-6, i=2,4,...,n When is odd, f*(waua2) = Hon) Sia) 2 5(2i-2)+1, i=1,3,5,... Hence the edges wj1u;2 carry labels 1,14, 21,--- ,10(n—1)+1 ifn is odd or 1,14, 21,--- , 10n—6 Mean Labelings on Product Graphs 101 if n is even. i i 1, Pua tee a) = eel C ne G2 1)2;-:2 n= 1 (since f(ui1) and f(uij41,1) are of opposite parity) = $15(24-1) +52¢+1)-2) +1] = 10:-2 Hence the edges uj1, Ui+i,1 have labels as 8,18, 28,--- ,10n — 12. f(ui2) + f(uiti,2) 2 (since f(ui2), f(uizi,2) are of same parity) = 10i—3, i=1,2,---,(n—1) f* (wa, Ui+1,2) The edges wiz, Uiz1,2 have 7,17,27,--- ,10n — 13 as labels. f (usa) + f(us41,3) 2 ie eae eee Ca f* (us, 41,3) = Therefore, uizui41,3 assume labels 10, 20, 30,--- ,10(n — 1), f(wia) + fuigia) +1 2 (since both vertex labels are of opposite parity) = 51502 1)+3+45(2i-1)+4+]] f*(wia, i414) = 10i-—1 1 or = 5 5(2i 3)+44+5(2i+1) +341) =10i-1 Therefore uj4ui+i,4 have labels as 9,19,--- ,10n — 11. When is odd, fr(uis, tg) = Lhwiad+ Fuss) I 2 5(2¢ — 2) +24+5(2i-1) +3 2 = 10i-—5 When 7 is even, f*(ujuia) = Luia) + Huis) sh — 10-1) +2+5(2i-3) +441 i a ea 101 — 9 I Hence 5,11, 25,---10n — 9 if n is even or 5,11,25,---,10n—5 if n is odd, correspond to the edges uj2uia f(uia) + f(uis) + 1 : f* (wia, wis) = 5 =107:-6+ (-1)’ 102 Teena Liza John and Mathew Varkey T.K So the edges u;2u;3 have labels 3,15, 23,--- ,10n —6+(-—1)”. f* (wiz, Uia) = Tuis) + Mus) = 10i —7 if i is even, or -_& fe) +? = 10) —4 if is oad So the values taken by uj3uj4 are 6,13,26,---10n —7 if n is even or 6,13, is odd. If ¢ is odd, f* (wa, wig) = LeadtiGdt’ _ io; 8 If 7 is even, f* (ui, uis) = Pun) fers) = 101-4 If 7 is odd, f* (wi, Uia) = Poin) fers) = 10i-6 If i is even, f* (wit, uia) = Poin) + Mus) = 10i-8 Hence the edge values of ujjusj are 1,2,4,--- ,10n — 8,10n — 6,10n 1,2,--- ,10n — 9,10n — 8,10n — 6 if n is odd. Hence the theorem. Example 3.1 The Fig.2 following shows the near mean labeling of PyOK4. Fig 2 ---,10n—4 ifn 4 if n is even, or Mean Labelings on Product Graphs 103 References 1] J.A.Gallian, The Electronic Journal of Combinatorics, 19 (2012), # DS6. 2| F.Harary, Graph Theory, Addison Wesley Publishing Company Inc. USA 1969. 3] A.Nagarajan, A.Nellai Murugan and S.Navaneetha Krishnan, On near mean graphs, I[n- ternational J. Math. Combin., Vol.4 (2010) 94-99 4] A.Nagarajan, A.Nellai Murugan and A.Subramanian, Near meanness on product graphs, Scientia Magna, Vol.6, No.3(2010), 40-49. 5] Richard Hammack, Wilfried Imrich and Sandi Klavzar, Hand Book on Product Graphs(2"4 edition), CRC Press, Taylor and Francis Group LLC, US, 2011. 6| S.Somasundaram and R.Ponraj, Mean labelings of graphs, National Academy Science Let- ter, 26 (2003) 210-213. Math. Combin. Book Ser. Vol.3(2014), 104-110 Total Near Equitable Domination in Graphs Ali Mohammed Sahal and Veena Mathad (Department of Studies in Mathematics, University of Mysore Manasagangotri, Mysore - 570 006, India) E-mail: alisahl1980@gmail.com, veena_mathad @rediffmail.com Abstract: Let G = (V,F) be a graph, D C V and wu be any vertex in D. Then the out degree of u with respect to D denoted by od, (u), is defined as od, (u) = |N(u)N(V — D)|. A subset D C V(G) is called a near equitable dominating set of G if for every v € V — D there exists a vertex u € D such that wu is adjacent to v and |od, (u) —od,,_ ,(v)| < 1. A near equitable dominating set D is said to be a total near equitable dominating set (tned-set) if every vertex w € V is adjacent to an element of D. The minimum cardinality of tned-set of G is called the total near equitable domination number of G and is denoted by yne(G). The maximum order of a partition of V into tned-sets is called the total near equitable domatic number of G and is denoted by dine(G). In this paper we initiate a study of these parameters. Key Words: Equitable domination number, near equitable domination number, near equitable domatic number, total near equitable domination Number, total near equitable domatic number, Smarandachely k-dominator coloring. AMS(2010): 05022 §1. Introduction By a graph G = (V,£) we mean a finite, undirected graph with neither loops nor multiple edges. The order and size of G are denoted by n and m, respectively. For graph theoretic terminology we refer to Chartrand and Lesnaik [2]. Let G = (V,£) be a graph and let v € V. The open neighborhood and the closed neigh- borhood of v are denoted by N(v) = {u € V: uv € E} and N[v] = N(v) U {v}, respectively. If SCV then N(S) = UvesN(v) and N[S] = N(S)US. Let G be a graph without isolated vertices. For an integer k > 1, a Smarandachely k- dominator coloring of G is a proper coloring of G with the extra property that every vertex in G properly dominates a k-color classes. Particularly, a subset S$ of V is called a dominating set if N[S] = V, ie., a Smarandachely 1-dominator set. The minimum (maximum) cardinality of a minimal dominating set of G is called the domination number (upper domination number) of G and is denoted by 7(G) ([(G)). An excellent treatment of the fundamentals of domination is given in the book by Haynes et al. [5]. A survey of several advanced topics in domination is given in the book edited by Haynes et al. [6]. Various types of domination have been defined and 1Received January 21, 2014, Accepted September 5, 2014. Total Near Equitable Domination in Graphs 105 studied by several authors and more than 75 models of domination are listed in the appendix of Haynes et al. [5]. E.J. Cockayne, R.M. Dawes and S.T. Hedetniemi [3] introduced the concept of total domination in graphs. A dominating set D of a graph G is a total dominating set if every vertex of V is adjacent to some vertex of D. The cardinality of a smallest total dominating set in a graph G is called the total domination number of G and is denoted by 7;(G). A double star is the tree obtained from two disjoint stars Ky, and Ky, by connecting their centers. Equitable domination has interesting application in the context of social networks. In a network, nodes with nearly equal capacity may interact with each other in a better way. In the society persons with nearly equal status, tend to be friendly. Let D C V(G) and wu be any vertex in D. The out degree of u with respect to D denoted by od, (wu), is defined as od,(u) = |N(u) MN (V — D)|. D is called near equitable dominating set of G if for every v € V — D there exists a vertex u € D such that u is adjacent to v and |od,, (wu) — od,_,(v)| <1. The minimum cardinality of such a dominating set is denoted by Yne and is called the near equitable domination number of G. A partition P = {Vi,V2,--- , Vi} ofa vertex set V(G) of a graph is called near equitable domatic partition of G if V; is near equitable dominating set for every 1 <i <J. The near equitable domatic number of G is the maximum cardinality of near equitable domatic partition of G and denoted by dye(G) [7]. For a near equitable dominating set D of G it is natural to look at how total D behaves. For example, for the cycle Cg = (v1, v2, U3, V4, U5, V6, U1), S1 = {1, va} and Sg = {v1, v2, v3, v4} are near equitable dominating sets, S; is not total and S3 is total. In this paper, we introduce the concept of a total near equitable domination to initiate a study of a total near equitable domination number and a total near equitable domatic number. We need the following to prove main results. Definition 1.1((7]) Let G = (V,E) be a graph and D be a near equitable dominating set of G. Then u € D is a near equitable pendant vertex if od, (u) =1. A set D is called a near equitable pendant set if every vertex in D is an equitable pendant vertex. Theorem 1.2([{7]) Let T be a wounded spider obtained from the star Kyn-1, n > 5 by subdi- viding m edges exactly once. Then n, iofm=n-1; Yne(T) = nw 1, ifm=n-—2; n—-2, ifm<n-3. §2. Total Near Equitable Domination in Graphs A near equitable dominating set D of a graph G is said to be a total near equitable dominating set (tned-set) if every vertex w € V is adjacent to an element of D. The minimum of the cardinality of tned-set of G is called a total near equitable domination number and is denoted by Yene(G). A subset D of V is a minimal tned-set if no proper subset of D is a tned-set. 106 Ali Mohammed Sahal and Veena Mathad We note that this parameter is only defined for graphs without isolated vertices and, since each total near equitable dominating set is a near equitable dominating set, we have Yne(G) < Yne(G). Since each total near equitable dominating set is a total dominating set, we have %(G) < yine(G). The bound is sharp for rK2, r > 1. In fact yYne(G) = %(G) = |VI, for G = rKg, it is easy to see however, that rK2, r > 1 is the only graph with this property. Furthermore, the difference yine(G) — 7%:(G) can be arbitrarily large in a graph G. It can be easily checked that y(41,-) = 2, while yine(Ki,,) =n — 2. We now proceed to compute ¥ne(G) for some standard graphs. 1. For any path P,, n > 4, n ; eae if n = 2 (mod 4); Yine(Pn) = (Pn) = =| otherwise 2° where [2] is a least integer not less than x. 2. For any cycle C,, n > 4, n ; aoe i if n = 2 (mod 4); =| , otherwise. 3. For the complete graph K,, n > 4 Yine(Kn) = Yne(Kn) = ISI where || is a greatest integer not exceeding «x. 4. For the double star Sym, 2 Yine(Sn,m) = “ne(Sn,m) = , 4 n+m-—2, ifn,m>2andnorm > 3. ifn,m<2; 5. For the complete bipartite graph Kym with 2 <m <n, we have m—-1, ifn=mandm> 3; Yine(Kn.m) = Yne(Kn.m) = Mm, ifn—m= 1; ifn—m> 2. n—-1, 6. For the wheel W,, on n vertices, vine(Wn) = Yne(Wn) = | Theorem 2.1 Let G be a graph and D be a minimum tned- set of G containing t near equitable n pendant vertices. Then y,,.(G) > a Total Near Equitable Domination in Graphs 107 Proof Let D be any minimum tned- set of G containing t near equitable pendant vertices t . Then 2|D| —t > |V — D]. It follows that, 3|D| —t >n. Hence ¥,,,.(G) > Theorem 2.2 Let T be a wounded spider obtained from the star Ky n-1, n > 5 by subdividing m edges exactly once. Then n, ifm=n—-1; Yine(T) = Yne(T) = n— 1; iofm=n—2; n—-2, ifm<n-3. Proof Proof follows from Theorem 1.2. Theorem 2.3 Let T be a tree of ordern, n > 4 in which every non-pendant vertex is either a support or adjacent to a support and every non- pendant vertex which is support is adjacent to at least two pendant vertices. Then Ytne(T) = Yne(T). Proof Let D denote set of all non-pendant vertices and all pendant vertices except two for each support of JT. Clearly, D is a Yne-set. Since any support vertex adjacent to at least two pendant vertices, it follows that (D) contains no isolate vertex. Hence D is a tned-set and hence Yine(T) < ne(T). Since Yne(T) < Yene(T), it follows that yine(T) = Yne(T). Theorem 2.4 Let G be a connected graph of order n, n > 4. Then, Yine(G) < n—2. Proof It is enough to show that for any minimum total near equitable dominating set D of G, |V — D| > 2. Since G is a connected graph of order n, n > 4, it follows that 6(G) > 1. Suppose vu € V — D and adjacent to u € D. Since od,,_,(v) > 1, then od, (wu) > 2. The star graph G = Kj, is an example of a connected graph for which Yine(G) = 2n — (A(G) + 3). The following theorem shows that, this is the best possible upper bound for yine(G). Theorem 2.5 If G is connected of order n, n > 4, then, Yene(G) < 2n — (A(G) + 3). Proof Let G be a connected graph of order n, n > 4, then by Theorem 2.4, yne(G) < n—2=2n—(n—1+83) < 2n—- (A(G) +3). Theorem 2.6 If G is a graph of ordern, n > 4 and A(G) < n—2 such that both G andG connected, then Yene(G) + Yine(G) < 3n-—6. Proof Let G be a connected graph and A(G) < n — 2. By Theorem 2.4, yne(G) < 2n — (A(G)+4) < 2n—(5(G)+4). Since G is a connected, by Theorem 2.5, yine(G) < 2n—(A(G)+3), 108 Ali Mohammed Sahal and Veena Mathad it follows that Yine(G) + Ytne (G) IA 9) Qn — (6(G) +4) + 2n — (A( An — (5(G) + A(G)) —7 ) +3) l| The bound is sharp for C4. Theorem 2.7 Let G be a graph such that both G and G connected. Then, Vtne(G) + Ytne(G) < 2n —A4. Proof Since both G and G are a connected, it follows by Theorem 2.4 that, yne(G) + Yine(G) < 2n — 4, The bound is sharp for Py. We now proceed to obtain a characterization of minimal tned-sets. Theorem 2.8 A tned- set D of a graph G is a minimal tned- set if and only if one of the following holds: (i) D is a minimal near equitable dominating set; (it) There exist x,y € D such that N(y) 1 N(D — {x}) = ¢. Proof Suppose that D is a minimal tned-set of G. Then for any u € D, D — {u} is not tned-set. If D is a minimal near equitable dominating set, then we are done. If not, then there exists a vertex x € D such that D— {x} is a near equitable dominating set, but not a tned- set. Therefore there exists a vertex y € D — {x} such that y is an isolated vertex in ((D — {x})). Hence N{y}N N(D — {x}) = 9. Conversely, let D be a tned- set and (7) holds. Suppose D is not a minimal tned-set. Then for every u € D, D — {u} is a tned- set. So, D is not a minimal near equitable dominating set, a contradiction. Next, suppose that D is a tned- set and (iz) holds. Then there exist x,y € D such that N(y) N N(D — {x}) =. Suppose to the contrary, D is not a minimal tned- set. Then for every u € D, D— {u} isa tned- set. So, ((D — {u})) does not contain any isolated vertex. Therefore for every x,y € D, N(y) A N(D — {x}) F ¢, a contradiction. Theorem 2.9 For any positive integer m, there exists a graph G such that ¥,,,.(G)— Fest = m, where |x| denotes the greatest integer not exceeding x. Proof For m=1, let G = K3,3. Then, 7,,.(G) — oI =2-1=1. For m = 2, let G = Ko4. Then, y,,,(G) — Fest =3-1=2. For m > 3, let G= S,,,, wherer+s =m+3,5s>r+3,r>2,7,,.(G) =r+s—-2=m+1, n _ r+s4+2 a4 A+1|~ | s+2 |] Total Near Equitable Domination in Graphs 109 and n Teele) ge Se OS Sm §3. Total Near Equitable Domatic Number The maximum order of a partition of the vertex set V of a graph G into dominating sets is called the domatic number of G and is denoted by d(G). For a survey of results on domatic number and their variants we refer to Zelinka [9]. In this section we present few basic results on the total near equitable domatic number of a graph. Let G be a graph without isolated vertices. A total near equitable domatic partition (tne- domatic partition) of G is a partition {Vi, V2,--- , Vz} of V(G) in which each V; is a tned-set of G. The maximum order of a tne-domatic partition of G is called the total near equitable domatic number (tne-domatic number) of G and is denoted by dine(G). We now proceed to compute dine(G) for some standard graphs. 1. For any complete graph Ky, n > 4, dine(Kn) = dne(Kn) = 2. 2. For any n > 1, dine(Can) = 2. 3. For any star Kin, n > 3, dime(Kin) = dne(Kiyn) = 1. 4. For the wheel W,, on n vertices, then dine(Wn) = dne(Wn) = 1. 5. For the complete bipartite graph Ky m, with 2<m<n 2 1 if |n—m| <2; dine(Kn,m) = dne(Kn.m) = ‘ if |n — m| > 3, n,m > 2. + It is obvious that any total near equitable domatic partition of a graph G is also a total domatic partition and any total domatic partition is also a domatic partition, thus we obtain the obvious bound dine(G) < di(G) < d(G). Remark 3.1 Let v € V(G) and deg(v) = 6. Since any tned-set of G must contain either v or a neighbour of v and dine(G) < d:(G), it follows that dine(G) < 6. Definition 3.2 A graph G is called tne-domatically full if dine(G) = 6. For example, a star Ky,,, is tne-domatically full. Remark 3.3 Since every member of any tne-domatic partition of a graph G on n vertices has at least yine(G) vertices, it follows that dine(G) < < —" _ This inequality can be strict for Ytne(G) rKo,r > 1. Theorem 3.4 Let G be a graph of ordern, n > 4 with A(G) < 2 such that both G and G are connected. Then dine(G) < 2. 110 Ali Mohammed Sahal and Veena Mathad proof Since A(G) < 2, it follows that for any v € G, deg(v) > n—3. Hence Yine(G) < [#]. Thus by Remark 3.3, dine(G) < 2. The bound is sharp for P,, n > 6. Theorem 3.5 Let G be a graph of order n, n > 4 with A(G) < 2 such that both G and G are connected. Then ine(G) + dine(G) <n. Proof Proof follows by Theorem 2.4 and Theorem 3.4. theorem 3.6 For any graph G, Yine(G@) + dine(G) < 2n — 3. proof By Theorem 2.5, Yine(G) < 2n — (A(G) +3) < 2n — (6(G) + 3) < 2n — (dine(G) + 3). Therefor, Yene(G) + dine(G) < 2n — 3. The bound is sharp for 2K9. theorem 3.7 For any graph G, Yine(G) + dine(G) <n+6—2. Proof Since dine(G) < di(G) < 6(G), by Theorem 2.4, Vtne(G) + dine(G) < nr + 6 =. 2. The bound is sharp for Ky. References 1] A.Anitha, S.Arumugam and Mustapha Chellali, Equitable domination in graphs, Discrete Mathematics, Algorithms and Applications, 3(2011), 311-321. 2] G.Chartrand and L.Lesnaik, Graphs and Digraphs, Chapman and Hall. CRC, 4th edition, 2005. 3] E.J.Cockayne, R.M.Dawes and S.T.Hedetniemi, Total domination in graphs, Networks, 10(1980), 211-219. 4| F.Harary and T.W. Haynes, Double domination in graphs, Ars Combin., 55(2000), 201-213. 5] T.W.Haynes, $.T.Hedetniemi and P.J.Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998. 6] T.W.Haynes, S.T.Hedetniemi and P.J.Slater, Domination in Graphs, Advanced Topics, Marcel Dekker, New York, 1998. 7| A.M.Sahal and V.Mathad, On near equitable domination in graphs, Asian Journal of Current Engineering and Maths., Vol.3, 2(2014), 39-46. 8] V.Swaminathan and K.Markandan Dharmalingam, Degree equitable domination on graphs, Kragujevac J. Math., 35(2011), 191-197. 9] B.Zelinka, Domatic number of graphs and their variants, in A Survey in Domination in Graphs Advanced Topics, Ed. T.W. Haynes, S.T.Hedetniemi and P.J.Slater, Marcel Dekker, 1998. 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Authors are advised to check their proofs very carefully before return. September, 2014 Contents Mathematics on Non-Mathematics — A Combinatorial Contribution BY LINFAN MAO On Cosets and Normal Subgroups BY B.O.ONASANYA AND S.A.ILORI On Radio Mean Number of Some Graphs BY R.PONRAJ AND S.SATHISH NARAYANAN Semientire Equitable Dominating Graphs BY B.BASAVANAGOUD, V.R.KULLI AND VIJAY V.TELI Friendly Index Sets and Friendly Index Numbers of Some Graphs BY PRADEEP G.BHAT AND DEVADAS NAYAK C Necessary Condition for Cubic Planar 3-Connected Graph to be Non-Hamiltonian with Proof of Barnette’s Conjecture BY MUSHTAQ AHMAD SHAH Odd Sequential Labeling of Some New Families of Graphs BY LEKHA BIJUKUMAR Mean Labelings on Product Graphs BY TEENA LIZA JOHN AND MATHEW VARKEY T.K Total Near Equitable Domination in Graphs BY ALI MOHAMMED SAHAL AND VEENA MATHAD An International Book Series on Mathematical Combinatorics