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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. 


It is also available from the below international databases: 


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Subscription A subscription can be ordered by an email directly to 
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Chinese Academy of Mathematics and System Science 
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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. 
G.B.Gurevich, Foundations of the Theory of Algebraic Invariants, P.Noordhoff Ltd. Gronin- 
gen, the Netherlands, 1964. 

David Hilbert, Theory of Algebraic Invariants, Cambridge University Press, 1993. 

H.Iseri, Smarandache Manifolds, American Research Press, Rehoboth, NM,2002. 

John M.Lee, Introduction to Topological Manifolds, Springer-Verlag New York, Inc., 2000. 
J.C.Lu, Fangfo Analyzing LAO ZHI - Explaining TAO TEH KING by TAI JI (in Chinese), 
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F.Klein, A comparative review of recent researches in geometry, Bull. New York Math. 
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Linfan Mao, On algebraic multi-group spaces, Scientia Magna, Vol.2, No.1 (2006), 64-70. 
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International J.Math. Combin. Vol.1(2007), No.1, 1-19. 

Linfan Mao, Geometrical theory on combinatorial manifolds, JP J.Geometry and Topology, 
Vol.7, No.1(2007),65-114. 

Linfan Mao, Combinatorial fields-an introduction, International J. Math.Combin., Vol.1(2009), 
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Linfan Mao, Relativity in combinatorial gravitational fields, Progress in Physics, Vol.3(2010), 
39-50. 


Linfan Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries, First 





edition published by American Research Press in 2005, Second edition is as a Graduate 
Textbook in Mathematics, Published by The Education Publisher Inc., USA, 2011. 
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by Hexis, Phoenix in 2006, Second edition is as a Graduate Textbook in Mathematics, 
Published by The Education Publisher Inc., USA, 2011. 

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lished by InfoQuest in 2005, Second edition is as a Graduate Textbook in Mathematics, 
Published by The Education Publisher Inc., USA, 2011. 

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bin., Vol.2 (2012), 9-23. 

Linfan Mao, Graph structure of manifolds listing, International J.Contemp. Math. Sci- 
ences, Vol.5, 2011, No.2,71-85. 


34 


24 


25 


26 


27 


28 











Linfan MAO 


Linfan Mao, A generalization of Seifert-Van Kampen theorem for fundamental groups, Far 
East Journal of Math.Sciences, Vol.61 No.2 (2012), 141-160. 

Linfan Mao, Global stability of non-solvable ordinary differential equations with applica- 
tions, International J.Math. Combin., Vol.1 (2013), 1-37. 

Linfan Mao, Non-solvable equation systems with graphs embedded in R”, International 
J.Math. Combin., Vol.2 (2013), 8-23. 

Linfan Mao, Non-solvable partial differential equations of first order with applications, 
submitted. 

F.Smarandache, Paradoxist Geometry, State Archives from Valcea, Rm. Valcea, Romania, 
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Probability and Statistics, American Research Press, 1998. 

F.Smarandache, A Unifying Field in Logics. Neutrosopy: Neturosophic Probability, Set, 
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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. 








Give me the greatest pleasure, not knowledge, but continuous learning; not 
for things, but constantly acquisition; not have reached the heights, but continued 
to clamb. 


By Johann Carl Friedrich Gauss, a Germany mathematician. 


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Books 


4\Linfan Mao, Combinatorial Geometry with Applications to Field Theory, InfoQuest Press, 
2009. 
12]W.S.Massey, Algebraic topology: an introduction, Springer-Verlag, New York 1977. 


Research papers 


6|Linfan Mao, Combinatorial speculation and combinatorial conjecture for mathematics, In- 
ternational J.Math. Combin., Vol.1, 1-19(2007). 
9|Kavita Srivastava, On singular H-closed extensions, Proc. Amer. Math. Soc. (to appear). 





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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