Edge Graceful Labeling of Some Trees
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1 Global Journal of Mathematical Sciences: Theory and Practical. olume, Number (0), pp. - International Research Publication House Edge Graceful Labeling of Some Trees B. Gayathri and M. Subbiah Lecturer (SG) in Mathematics, Periyar ER College, Trichy 0 0, India maduraigayathri@gmail.com Abstract Lo [] introduced the notion of edge graceful graphs : A graph G with q edges and p vertices is said to be edge graceful if there exists a bijection f from the edge set to the set {,,..q} so that the induced mapping f + from the vertex set to the set {0,,,..p-} given by f + (x) = {f (xy) / xy E (G)} (mod p) is a bijection. In this paper, we investigate edge graceful labeling of bistar graph. Introduction All graphs in this paper are finite, simple and undirected. Terms not defined here are used in the sense of Harary []. The symbols (G) and E(G) will denote the vertex set and edge set of a graph G. The cardinality of the edge is called the size of G. A graph with p vertices and q edges is called a (p, q) graph. A labeling (or valuations) [] of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy, a label depending on the vertex labels f (x) and f (y).let G be a graph with q edges. Let f be an injection from the vertices of G to set {0,,,.q} is called a graceful labeling of G if when we assign to each edge xy the label f(x) f (y) the resulting edge labels are distinct. Lo [] introduced the notion of edge graceful graphs. A graph G with q edges and p vertices is said to be edge graceful if there exists a bijection f from the edge set to the set {,,.q} so that the induced mapping the vertex set to the set {0,,.p-} given by f + (x) = {f (xy/xy E (G)} (mod p) is a bijection. The necessary condition for a graph to be edge graceful is q (q+) 0 or p/ (mod p). With this condition one can vertify that even cycles, and paths of even length are not edge graceful. But whether trees of odd order are edge graceful is still open. On attempting to move towards this conjecture, in this paper we checked it for a special type of trees called bistar graph. Here again the odd order trees turn to be edge graceful graph. Thus it confirms that the conjecture is moving towards affirmative.
2 B. Gayathri and M. Subbiah Main Results Definition:. The graph B n, m is defined as the graph obtained by joining the center u of the star K, n and the center of another star K, m to a new vertex w. Observation:. The number of vertices in the graph B n, m is p = n+m+ and the number of edges q = n+m+. Theorem:. The bistar graph B n, m for n m where n, m even is an edge graceful graph. Let {w, v,,,.v n, v n+ v n+m } be the vertices of bistar and edges e i (see fig ) are defined as follows n v n+m n- n- e n- e n- e e n e w e n+m e n+m- v n+m- e n+ e n+ v n+ v n+ Figure: B n, m with ordinary labeling. e = (w, v); = (w, ); e i = (v, v i ) for i =,, n and e i = (, v) for i = n+, n+, n + m. Consider the Diophantine equation x +x = p. The solutions are of the form (t, p-t) wher t q/. There will be q pair of solutions. With these pairs label the edges of K, n and K, m by the coordinates of the pair in any order so that adjacent edges receives the coordinates of the pairs. Also, label f + (w) = 0 ;f + ( ) = ; f + (v) = q. Now the pendant vertices will have labels of the edges with which they are incident and they are distinct. Hence the graph B n, m & n m n, m even is edge graceful. Consider the bistar graph B,. Here p = 9
3 Edge Graceful Labeling of Some Trees 9 v e e e e w e v Figure: Edge graceful labeling of B,. Consider the Diophantine equation x +x = p =9 The pair of solutions are : (t, p-t) where t (, q/) : (, ), (, ), (, ), (, ). We label the edges e & as follows: f (e) = q =, f( ) =. Pair of solutions labels the edges of K, & K, by the coordinate of the pairs. The edge graceful labeling of bistar B, is given in fig. 0 Figure: Edge graceful labeling of B,. Theorem:. The bistar graph B n, m for n m where n, m odd is a edge graceful graph. Consider the ordinary labeling of the vertices & edges of B n, m as was done in Theorem.. Consider the Diophantine equation x +x = p. The solutions are of the form (t, p-t) wher t q/. There will be q pair of solutions. Among q pair of solutions, choose the pair (q, ). We now label the edges as follows, f (e) = q ;f ( ) = ; f ( ) = ; f (e n+ ) = p. As we have labeled the edges and e n+ there will be even number of edges incident with each v and to be
4 0 B. Gayathri and M. Subbiah labeled. We label these edges as was done in Theorem.. Then the induced vertex labels are f + (w) = 0; f + (v) = q + ; f + ( ) = p-. The pendant vertices will have the labels of the edges with which they are incident. They are distinct. Hence the graph bistarb n,m for n m, n, m odd is an edge graceful graph. Consider the bistar graph B,. Here p =, q = 0. v v e v e e e v e e w e v v Figure: B, with ordinary labeling Figure: Edge graceful labeling of B,. Consider the Diophantine equation x + x =. The pair of solutions are: (, 0), (, 9), (, ), (, )and (,). Let us choose the pair(q,)=(,0) Define f (e) = q; f( ) =. From the remaining pair label the edges as discussed in Theorem.. The edge graceful labeling of bistar B, is given in Fig.. Theorem:. The bistar graph B n, m for n = m where n, m even is an edge graceful graph. Consider the ordinary labeling of the vertices and edges of B n, m as was done in Theorem.. Consider the Diophantine equation, x + x = p. The solutions are of the form (t, p- t) wher t q. There will be q pair of solutions. With these pairs label the edges of K, n and K, m by the coordinates of the pair in any order so that adjacent edges receives the coordinates of the pairs. Also, label f(e)
5 Edge Graceful Labeling of Some Trees = q; f ( ) =. Then the induced vertex labels are f + (w) = 0; f + ( ) = ; f + (v) = q. Now the pendant vertices will have labels of the edges with which they are incident and they are distinct. Hence the graph B n, m & n = m even is edge graceful. Consider the bistar graph B,. Here p =, q = consider the Diophantine equation x +x =. The pair of solutions are (t, p-t) where t (, q/) (, ), (, ), (, ). We label the edges e & as follows f(e) = q = ; f( ) = we use the pair of solutions to label the edges of K, &K, and we label them by the coordinate of the pairs. The edge graceful labeling of bistar B, is given in fig. v e e w e v Figure: B, with ordinary labeling. 0 Figure9: Edge graceful labeling of B,. Theorem:. The bistar graph B n, m for n = m where n, m odd is an edge graceful graph. Consider the ordinary labeling of the vertices and edges of B n, m as was done in Theorem.. Consider the Diophantine equation x +x =p. The solutions are of the form (t,p-t) where t q/. There will be q/ pair of solutions. Among q pair of solutions, choose the pair (q, ). We now label the edges as follows, f(e) = q; f ( ) = ; f ( ) = ; f (e n+ ) = p-. As we have labeled the edges and e n+ there will be even number of edges incident with each v and to be labeled. We label these edges as was done in Theorem.. Then the induced vertex labels are f + (w) = 0; f + (v) = q + f + ( )=p-. The pendant vertices will have the labels of the edges with which they are incident. They are distinct. Hence the graph bistar B n,m, n,m odd is an edge graceful graph.
6 B. Gayathri and M. Subbiah Consider the bistar graph B,. Here p = 9; q = v e e v e e w e v v Figure0: B, with ordinary labeling. 0 Figure: Edge graceful labeling of B,. Consider the Diophantine equation x +x = 9. The pair of solutions are; (,), (, ), (, ), (, ). Let us choose the pair (, q) = (, ) label the edges as follows f(e) = q; f( )=. From the remaining pairs label the edges as discussed in Theorem.. The edge graceful labeling of bistar B, is given in Fig. Theorem:. [Lo] [] If a graph is edge graceful then q (q+) p( p ) (mod p) Theorem:. Any tree of even order is not edge graceful By Theorem. Theorem:.9 The graph bistar B n, m of even order is not an edge graceful graph. The graph B n, m will be even order in the following cases.. n is odd and m is even. n is even and m is odd In both cases p = n + m +, which is even and hence by Theorem. it is not an edge graceful.
7 Edge Graceful Labeling of Some Trees Reference [] Harary. F Graph Theory Addison wesley mass reading (9) [] Gallian J.A A dynamic survey of graph labeling The electronic journal of combinatorics (00) t DS#. [] Gayathri. B and Subbiah. S - Strong edge graceful labeling of some graphs presented in the NCAMPER (00) Periyar ER College, Trichy-, March -. [] Gayathri. B and Subbiah. S - Strong edge graceful labeling of some trees presented in the National Conference at Jamal Mohamed College, Trichy On March - (00). [] Strong Edge graceful Labeling for some graphs accepted for publication in Bulletin of pure and Applied Science, vol..no.,00 to appear. [] Lo.S on edge graceful labeling of graphs, Congressus Numerantium 0 (9). [] Slamet. S and suging K.A sharing scheme using magic covering preprint.
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