Karen L. Collins. Wesleyan University. Middletown, CT and. Mark Hovey MIT. Cambridge, MA Abstract

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1 Mot Graph are Edge-Cordial Karen L. Collin Dept. of Mathematic Weleyan Univerity Middletown, CT 6457 and Mark Hovey Dept. of Mathematic MIT Cambridge, MA 239 Abtract We extend the definition of edge-cordial graph due to Ng and Lee for graph on 4k, 4k +, and 4k +3 vertice to include graph on 4k +2 vertice, and how that, in fact, all graph without iolated vertice are edge-cordial. Ng and Lee conjectured that all tree on 4k, 4k +,or 4k + 3 vertice are edge-cordial. Intuitively peaking, a graph G i aid to be edge-cordial if it edge can be labelled either or o that half of the edge are labelled ; half are labelled ; and half of the vertice meet an even number of edge labelled, while the other half meet an odd number of edge labelled. Thi definition i due to Ng and Lee, ee [2], and i the dual concept to cordial graph, firt introduced by Cahit in []. Ng and Lee howed that if the number of vertice in any graph i congruent to 2 (modulo 4), then the graph cannot be edge-cordial. We extend their definition of edge-cordial to graph on 4k + 2 vertice. Definition Suppoe we have a imple graph G =(V; E). Let f : E! f; g be anedge-labelling of G. Define a vertex-labelling f Λ : V!f; g by f Λ (v) = X (u;v)2e f(u; x) (mod 2) Let E i denote the et of edge labelled i, and V i be the et of vertice labelled i, for i =;. Then G i aid to be edge-cordial if and only if. (Ng and Lee) j V j ; ; 3 (mod 4) and there exit an edge-labelling f uch that j (j E j je j) j» and j (j V j jv j) j» ; or

2 2. j V j 2 (mod 4), and there exit an edge-labelling f uch that j (j E j je j) j» and j (j V j jv j) j =2: Conjecture (Ng and Lee) All tree on 4k, 4k +, or 4k +3 vertice are edge-cordial. Graph with 4k+2 vertice cannot atify the firt condition in the definition becaue of the following lemma. Lemma (Ng and Lee) j V j i even. Proof Conider the ubgraph H of G induced by the edge labelled. The vertice labelled in G are the vertice of odd degree in H. Clearly there mut be an even number of vertice of odd degree in H. In thi paper, we prove that if G i a graph with no iolated vertice then G i edge-cordial. In particular, Ng' and Lee' conjecture i true. Theorem Tree are edge-cordial. Proof It i eay to check that tree with 2; 3or4vertice are edge-cordial. We proceed by induction. Suppoe T i a tree with 5 vertice and uppoe that all tree with j V j 4 vertice are edge-cordial. We how that T i edge-cordial. Firt aume that T ha two leave, v and v 2, both adjacent tow. If T ha at leat four leave, we can remove v and v 2 and two other leave, u and u 2, (and their repective edge) and label the remaining graph by induction. We extend the labelling to T by coloring the edge incident with v and v 2 both and the other two edge both. Thi add 2 vertice that ee an odd number of one, and 2 that ee an even number, without changing the parity of previouly labelled vertice. The vertice u and u 2 may be adjacent tow or not. If T ha le than 4 leave, then it ha either 2 or 3 leave. If T ha only 2 leave, then T cannot have at leat 5 vertice. 2

3 If T ha 3 leave, then T minu v and v 2 mut be a path, P. If P ha at leat 4 vertice we can remove 4 of them and color them and complete the coloring by induction. If P contain < 4 vertice, then (ince T ha at leat 5 vertice) T i which i eaily labelled a hown. Now uppoe that T doe not have 2 leave both adjacent to the ame vertex. Let l, l 2 be 2 (vertex) leave which are maximally ditant int. Let u and u 2, repectively, be their neighbor. We will how that the degree of u (and hence, by ymmetry, the degree of u 2 )i2. Obviouly the degree of u 2 ince u mut have an edge e included in the path between l and l 2. Suppoe u ha another edge d connecting u to v. Then v mut be a leaf, ince l and l 2 are maximally ditant. But then l and v are both leave and both adjacent tou. So degree of u = 2. Let w be the other end of e. Let e 2 be the edge adjacent tow but not u in the path between l and l 2. Suppoe w ha another edge e 3 connecting it to a vertex v. Any edge from v other than e 3 mut end in leave ince l and l 2 are maximally ditant. Since T ha no two leave both adjacent to the ame vertex, there can be at mot one edge from v other that e 3,ay e 4. Let the other end of e 4 be t. Ife 4 exit, T look like e 2 l2 e l u w e u 3 2 v e 4 t But then T fl ;u ;v;tg ha an edge-cordial labelling, and we can complete that labelling a hown: 3

4 Hence we can aume that e 4 doe not exit, and that v i a leaf. There cannot be another edge beide e ;e 2 ;e 3 coming off from w, ince, uing the ame argument, we could conclude that it lead to a leaf, and we would have 2 leave both adjacent to w. So T look like: and we can complete an edge-cordial labelling a hown. The remaining cae i when the degree of w i 2, and T look like l u w u 2 l2 In thi cae we can complete an edge-cordial labelling of T fl ;u ;w;l 2 g a hown. So by induction T i edge-cordial. We can ue thi reult to prove Theorem 2 Foret with no iolated vertice are edge-cordial. Proof Let F be a foret with connected component F ;F 2 ;:::;F k. Each F i i a tree. If one of the F i ha at leat j V i j 6vertice, we can ue the ame argument that we ued in the induction argument for tree to reduce the number of vertice in F i to j V i j 4. Since 6 4 = 2, thi new tree will not be an iolated vertex. (Thi i becaue we tookaway the ame number of edge a edge, and didn't change any vertex label. Alo, we left a connected graph.) Thu we need only conider foret F with each F i having 2, 3, 4, or 5 vertice. If there are two 2vertex tree, we can color one edge and the other. Then we get two vertice and two vertice, o we can reduce thi cae. If there are two tree with 4 vertice, we can ue a coloring on one with 2 zero edge and one edge, and on the other, 2 one edge and zero edge. One of the four cae of thi i given below. 4

5 Uing imilar technique (of matching a mall tree with more zero edge to one with more one edge) we can reduce the following cae: a 3 vertex tree and a 5 vertex tree; four 3 vertex tree; four 5 vertex tree. Thu we have reduced to a bounded numberofeach kind of tree and one can jut check all poible graph. Suppoe we have an edge-cordial graph G with more edge than edge. We would like tochange the labelling o that we have more edge than edge. Below we lit ome ituation in which we can do thi. We will how a piece of G together with the change we make on that piece to fix the labelling. Situation ) Situation 2 ) Situation 3 k edge A k + -cycle of edge ) k edge A k + -cycle of edge Situation 4 ) 5

6 Situation 5 ) Situation 6 ) Situation 7 ) Each of thee ituation ha a dual obtained by interchanging the vertex labelling. If we et V! V and V! V, then we obtain a new ituation which alo work. Situation, 6, and 7 are elf-dual. Theorem 3 If G i an edge-cordial graph with no iolated vertice and an odd number of edge, then there i an edge-cordial labelling of G with j E j j E j Proof Chooe an edge-cordial labelling of G. Suppoe with thi labelling j E j>j E j. By ituation above wemay aume that there are no edge between oppoitely labelled vertice. Thu there mut be either a edge between two vertice or a edge between two vertice. Cae : The only edge are between two vertice. If j V j 2 (mod 4), we can aume j V j<j V j, ince otherwie we could change a edge between two vertice to a edge and till have an edge-cordial labelling. Now there mut be two adjacent edge. If not, j E j»j V j =2 o j E j<j V j =2 and G would have tohave an iolated vertex. Hence by Situation 2, we may aume there are no edge labelled between two vertice. Thu the only edge adjacent to vertice are labelled and connect a vertex to a vertex. Let k denote the numberofuch edge. 6

7 Since G ha no iolated vertice k j V j (j V j )=2: Hence by Situation 3, we may aume that there are no cycle of edge of ize» (j V j +)=2: But there are at mot (j V j +)=2 vertice labelled. Thu the ubgraph ofvertice and edge form a foret. So we have j V j»j E j»j E j»j V j which violate the edge-cordiality of G. Cae 2: The only edge are between two vertice. Thi cae i the ame a the previou cae. One jut ue the dual of the ituation ued above. Cae 3: Both kind of edge appear. If j V j 2 (mod 4), then either j V j = j V j +2or j V j = j V j +2 In either cae we can change a edge to a edge and till have an edgecordial labelling. So we aume j V j6 2 (mod 4). Since we have both kind of edge we can aume, by Situation 4 and it dual, that if a edge and a edge are adjacent the edge mut have oppoitely labelled vertice. Suppoe the edge between vertice do not form a matching. Take two adjacent one. They form a two-edge path. The endpoint of thi path are labelled o they mut be adjacent to a edge. The only poibility i a edge between oppoitely labelled vertice. Thu we are in the dual of Situation 5 and we can reduce. So we can aume that the edge between vertice form a matching. Suppoe there i more than one uch edge. Take a vertex adjacent to one of them. Again ince it i labelled it i alo adjacent to a edge between oppoitely labelled vertice. If the vertex it i adjacent to ha a edge coming off it we are in Situation 6. On the other hand if that vertex doe not have a edge coming off it we are in Situation 7. Hence we can aume there i only one edge labelled between two vertice. We know that there are two edge labelled, one for each endpoint of that edge. Let l denote the number of other edge. Let m denote the number of edge between two vertice. Then l +2 =j E j<j E j= m + o 2l» 2m 4. Now if the edge form a matching we have j V j» 2l +2< 2m»j V j which violate edge-cordiality. Thu the edge do not form a matching and by Situation 2 we may aume there are no edge labelled between vertice. Since every 7

8 vertex mut be adjacent to a edge there are at leat j V j edge between oppoitely labelled vertice. Thu by Situation 3, there are no cycle of ize le than or equal to j V j + of edge between vertice. Since there are at mot j V j + vertice labelled, the edge form a foret. If thi foret doe not cover all the vertice we have j V j»j E j<j E j» (j V j 2) + = j V j contradicting edge-cordiality. Thu every vertex i adjacent to a edge. Conider the edge between two vertice. An endpoint of thi edge mut have a edge connecting it to avertex coming off it. Thi vertex i connected to another vetex. Thu we are in Situation 6 and we can reduce. The lat two theorem immediately imply Theorem 4 Graph with no iolated vertice are edge-cordial. Proof We proceed by induction on the number of edge attached to a panning foret. The bae cae i Theorem 2. Suppoe all foret with k extra edge are edge-cordial. If G i a foret with k + extra edge, delete an edge that i not in a panning foret. By induction thi graph i edge-cordial. By Theorem 3 we can find an edge-cordial labelling with j E j»j E j. Label the new edge. Then thi labelling i edge-cordial. Reference [] I. Cahit, Cordial graph: a weaker verion of graceful and harmoniou graph, Ar. Comb (987). [2] H. K. Ng and S. M. Lee, A conjecture on edge-cordial tree, Abtract Amer. Math. Soc (988). 8