INTERSECTION CORDIAL LABELING OF GRAPHS

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1 INTERSECTION CORDIAL LABELING OF GRAPHS G Meea, K Nagaraja Departmet of Mathematics, PSR Egieerig College, Sivakasi- 66 4, Virudhuagar(Dist) Tamil Nadu, INDIA meeag9@yahoocoi Departmet of Mathematics, Kalasaligam Uiversity, Aad Nagar, Krishakovil , Srivilliputhur(via), Virudhuagar(Dist) Tamil Nadu, INDIA, k agaraja srmc@yahoocoi Abstract - A itersectio cordial labelig of a graph G with vertex set V is a ijectio f from V to the power set of {,,, } such that if each edge uv is assiged the label if f (u) f (v) 6 ad otherwise; The the umber of edges labeled with ad the umber of edges labeled with differ by at most If a graph has a itersectio cordial labelig, the it is called itersectio cordial graph I this paper, we proved the stadard graphs such as path, cycle, wheel, star ad some complete bipartite graphs are itersectio We also proved that complete graph is ot itersectio Key words: Cordial labelig, Itersectio cordial labelig, Itersectio cordial graphs Graph labelig [4] is a strog commuicatio betwee Algebra [] ad structure of graphs [5] By combiig the set theory cocept i Algebra ad Cordial labelig cocept i Graph labelig, we itroduce a ew cocept called itersectio cordial labelig A vertex labelig [4] of a graph G is a assigmet f of labels to the vertices of G that iduces for each edge uv a label depedig o the vertex label f (u) ad f (v) The two best kow labelig methods are called graceful ad harmoious labelig Cordial labelig is a variatio of both graceful ad harmoious labelig [] Defiitio [] Let G (V, E) be a graph A mappig f : V (G) {, } is called biary vertex labelig of G ad f (v) is called the label of the vertex v of G uder f INTRODUCTION For a edge e uv, the iduced edge labelig f : E(G) {, } is give by f (e) f (u) f (v) Let vf () ad vf () be the umber of vertices of G havig labels ad respectively uder f ad ef () ad ef () be the umber of edges havig labels ad respectively uder f The cocept of cordial labelig was itroduced by Cahit [] By a graph, we mea a fiite, udirected graph without loops ad multiple edges, for terms ot defied here, we refer to Harary [5] Here G deotes (p, q) graph, where p is the umber of vertices ad q is the umber of edges of G First we give the set cocepts i Algebra [] Let X {,,, } be a set ad (X) be the collectio of all subsets of X, called power set of X If A is a subset of B, we deote it by A B otherwise by A 6 B Note that (X) cotais subsets Let Cr deotes the umber of ways of selectig r objects from objects Defiitio [] A biary vertex labelig of a graph G is called a cordial labelig if vf () vf () ad ef () ef () A graph G is cordial if it admits cordial labelig Cahit proved some results i [] ISSN: -57 Page 67

2 Mai Results Here ef () 4 ad ef () ef () ef () Thus G is itersectio DK Natha ad KNagaraja have itroduced subset cordial labelig ad they have proved that some graphs are subset cordial [6] Example 6 I the example 4, by removig the vertices labeled {, } ad, we get the followig itersectio cordial graph {} {} {,} Example 4 Cosider the followig graph G Take X {,, } {} {,} {,,} {,} {,} {,} Proof Let X {,,, } Assume that p Let vp be the cetral vertex ad v, v,, vp be the ed vertices of the star Note that, K,q cotaiig q edges First, we label the cetral vertex by ay -elemet set, say {} Label the remaiig ed vertices by the other subsets of X We observe that the itersectio of each subset of {,, 4,, } with {} is ad so it cotributes, s to ef () The remaiig subsets cotribute, s to ef () Thus ef () ef () ad so the star graph K,q is itersectio {,,} Theorem 7 The star graph K,q is itersectio Here ef () ad ef (), ef () ef () Thus G is itersectio {} I this paper, we prove the stadard graphs such as path, cycle, star, wheel ad some complete bipartite graphs are itersectio We also prove that complete graph is ot a itersectio First we prove that star graphs are itersectio Example Cosider the followig graph G Take X {, } {,} Defiitio Let X {,,, } be a set Let G (V, E) be a simple (p, q) graph ad f : V (X) be a ijectio Also, let < p For each edge uv, assig label if either f (u) f (v) 6 ad if f (u) f (v) f is called a itersectio cordial labelig if ef () ef () A graph is called a itersectio cordial graph if it has a itersectio cordial labelig {} Motivated by the cocept of subset cordial labelig, we itroduce a ew special type of cordial labelig called itersectio cordial labelig as follows 5, Remark 5 If < p <, we take X {,,, }, the we have more umber of subsets tha vertices So we ca easily label the vertices so that ef () ef () So, for provig itersectio cordiality of graph, it eough to prove for p Defiitio [6] Let X {,,, } be a set Let G (V, E) be a simple (p, q) graph ad f : V (X) be a ijectio Also, let < p For each edge uv, assig label if either f (u) f (v) or f (v) f (u) or assig if f (u) is ot a subset of f (v) ad f (v) is ot a subset of f (u) f is called a subset cordial labelig if ef () ef () A graph is called a subset cordial graph if it has a subset cordial labelig {} The labelig patter give i the Theorem 7 is illustrated i the followig example Example 8 Cosider the followig star graph G Take X {,,, 4} ISSN: -57 Page 68

3 {,,4} {,,4} {,,4} {,,,4} {,4} {4} ef () {} [ k + ] k [ + ] [( + ) ] {} {,,} Hece {,4} {,} {,} {,} {,4} We have ef () + ef () q The p(p ) p(p ) ef () ( ) ( ) ef () Here ef () 8 ad ef () 7 Thus ef () ef () ad hece G is itersectio Next, the itersectio of cordiality of complete graphs are discussed It is clear that Kp is itersectio cordial for p, ad We will prove that Kp is ot itersectio cordial for p 4 Now ( ) + ( ) ( ) for ef () ef () Theorem 9 K is ot itersectio cordial for Thus K is ot itersectio Next, we prove that path is itersectio Proof Let X {,,,, } ad p Label the vertices of K by all subsets of X We first calculate ef () Note that the itersectio of the empty set with ay subset of X is ad it cotributes to ef () Now cosider the sigleto sets For example, take {} The umber of subsets of the set {, } is We see that the itersectio of the subsets of {} with ay subset of {,,,, } is ad the empty subset has already be take i accout So, it cotributes to ef () Sice the umber of sigleto sets is, all the sigleto sets cotribute to ef () ad so ef () Next, we cosider the two elemets sets For example, take {, } The umber of subsets of the set {, 4,, } is The itersectio of the set {, } with every subset of {, 4,, } is ad so it cotributes to ef () There are two elemets sets, ad so these two elemets subsets cotribute to ef () three elemets subsets Similarly, cotribute to ef () I geeral, all k the k -members subsets cotribute k to ef () I a complete graph, we observe that each cotributio couted as twice Theorem The path P is itersectio Proof Let X {,,, } ad let p The q Let V (P ) {v, v, v } Now, we arrage all the subsets of X i the followig patter Label the first vertex of P by ad the secod vertex of P by X Next, label the third vertex of P by -elemet set ad fourth vertex by its complemet Cotiue this process util, the elemet subsets are exhausted After that we label the -elemet subset ad that ext by its complemet Cotiue this process, util the -elemet subsets are exhausted Now, cotiuig by labels of -elemet, 4-elemet subsets ad this process will ed elemet subset if is eve The k upto elemet subsets were (k> ) are already la beled ad so the labelig is completed Thus all the subsets of X are exhausted If is odd, we cotiue the above process - elemet subsets upto Now, we see the labelig of edges of P Clearly the labelig of first edge i P is Note that the itersectio of X ad the elemet set is o empty, so the labelig of secod edge is Next we labeled the vertices by the set ad its complemet alteratively ISSN: -57 Page 69

4 The labelig of edges icidet with u are ad icidet with u are, sice the itersectio of with ay set is ad itersectio of whole set X with ay subset Y of X is Y Thus, we see that ef () ef () Hece K, is itersectio ad so the labels ad s are labeled alteratively to the edges of P Thus we have ef () ad ef () ad so ef () ef () Hece P is itersectio Example Cosider the path P P8 Example 5 Cosider the graph K,6 {,,}{} {,} {} {,} {,} {,,} Here ef () 4; e f () Thus ef () ef () Theorem The cycle C is itersectio {} Theorem 6 {,4} {,,,4} {} {,4} {,,4} {,} {} {,4} {,} {,,} {,,4} {4} K, is itersectio No i) ii) Edges icidet with u u iii) u Total {,,4} 4 ef () 5 Example 7 Cosider the followig complete bipartite graph K,5 Here ef () 8, ef () 8 Thus ef () ef () Now, we will prove some complete bipartite graphs are itersectio {,,} φ Theorem 4 ef () Thus ef () ef () ad so K, is itersectio {,} Proof Note that K, has vertices ad 9 edges Let V (K, ) A B where A {u, u, u } ad B {v, v,, v } Cosider the set X {,,,, } Now we label all the subsets of X to the vertices K, as follows First we label the vertex u by, u by the whole set X ad u by ay -elemet subset, say {} Next we label the remaiig subsets of X to the vertices v, v,, v The labelig of edges icidet with u, u ad u are give i the followig table Example Cosider the cycle C6 {,} {,} Here ef () 6 ad ef () 6 Thus the complete bipartite graph K,6 is itersectio Proof Cosider the labelig of path P i the Theorem Now joiig the iitial ad fiial vertices of P we get the cycle C ad the ew edge joiig iitial ad fiial vertices of path get the label We see that ef () + ad ef () But ef () ef () To make itersectio cordiality, we iterchage the labelig of vertices already labeled by {,,, } ad {} Now we get ef () ef () ad ef () ef () So C is itersectio {,} {} K, is itersectio Proof Let V (K, ) A B, where A {u, u } ad B {v, v,, v } Cosider the set X {,,, } Now we label all the subsets of X to the vertices of K, as follows First, we label the vertex u by ad u by the whole set X Next we label the remaiig subsets of X to the vertices v, v,, v {} {} {,} {,} {,} Here ef () 8 ad ef () 7, ad ef () ef () Thus, the complete bipartite graph K,5 is itersectio 4 ISSN: -57 Page 7

5 Refereces Coclusio 8 I this paper, we have checked itersectio cordiality of some stadard graphs Further, we are tryig to establish the itersectio cordiality of the followig various structure of graphs (i) Subdivisio of stadard graphs (ii) Wheel related graphs (iii) Cycle related graphs (iv) Star related graphs (v) Coroa of graphs (vi) Trasformatio graphs [] I Cahit, Cordial graphs: A weaker versio of graceful ad harmoious graphs, Ars combiatoria, (987), -7 [] I Cahit, O cordial ad -equitable labeligs of graph, Utilitas Math, 7(99), [] INHerstie, Topics i Algebra, Secod Editio Joh Wiley ad Sos, 999 [4] J A Gallia, A dyamic survey of graph labelig, Electroic Joural of Combiatorics, 6(9), DS6 Also, we are goig to establish the relatioship betwee itersectio cordiality ad other cordiality such as subset cordiality, divisor cordiality, prime cordiality etc [5] F Harary, Graph Theory, Addiso-Wesley, Readig, Mass, 97 [6] DK Natha ad KNagaraja, Subset Cordial Graphs, Iteratioal Joural of Mathematical Scieces ad Egieerig Applicatios Vol7(Sep-) 5 ISSN: -57 Page 7