Difference Cordial Labeling of Graphs Obtained from Double Snakes

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1 International Journal of Mathematics Research. ISSN Volume 5, Number 3 (013), pp International Research Publication House Difference Cordial Labeling of Graphs Obtained from Double Snakes R. Ponraj, S. Sathish Narayanan and R. Kala Department of Mathematics, Sri Paramakalyani College, Alwarkurichi-6741, India. ponrajmaths@gmail.com Department of Mathematics, Thiruvalluvar College, Papanasam-6745, India. sathishrvss@gmail.com Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-6701, India. karthipyi91@yahoo.co.in Abstract Let G be a (p, q) graph. Let f: V(G) {1,,, p} be a function. For each edge uv, assign the label f(u) f(v). f is called a difference cordial if f is a one to one map and e (0) e (1) 1 where e (1) and e (0) denote the number of edges labeled with 1 and not labeled with 1 respectively. A graph with admits a difference cordial labeling is called a difference cordial graph. In this paper, we investigate the difference cordial labeling behavior of several graphs which are obtained from triangular snake and quadrilateral snake. Keywords: corona, comb, tree, cycle, wheel. 010 Subject Classification: 05C78 Introduction: Throughout this paper we have considered only simple and undirected graph. Let G = (V, E) be a (p, q) graph. The cardinality of V is called the order of G and the cardinality of E is called the size of G. Labeled graphs are used in several areas such as astronomy, radar and circuit design and database management [1]. The corona of G with H, G H is the graph obtained by taking one copy of G and p copies of H and joining the i th vertex of G with an edge to every vertex in the i th copy of H. The concept of difference cordial labeling has been introduced by R. Ponraj, S. Sathish

2 318 R. Ponraj, S. Sathish Narayanan and R. Kala Narayanan, R. Kala in [3]. In [3, 4, 5] difference cordial labeling behavior of several graphs like path, cycle, complete graph, complete bipartite graph, bistar, wheel, web, some snake graphs, crown C K, comb P K, P C, C C, W K, W K and some more standard graphs have been investigated. In this paper we investigate the difference cordial labeling behavior of DT K, DT K, DT K, DQ K, DQ K, DQ K where DT and DQ are double triangular snake and double quadrilateral snake respectively. Let x be any real number. Then x stands for the largest integer less than or equal to x and x stands for the smallest integer greater than or equal to x. Terms and definitions not defined here are used in the sense of Harary []. Difference Cordial Labeling Definition.1: Let G be a (p, q) graph. Let f be a map from V(G) to {1,,, p}. For each edge uv, assign the label f(u) f(v). f is called difference cordial labeling if f is 1 1 and e (0) e (1) 1 where e (1) and e (0) denote the number of edges labeled with 1 and not labeled with 1 respectively. A graph with a difference cordial labeling is called a difference cordial graph. A double triangular snake DT consists of two triangular snakes that have a common path. That is, a double triangular snake is obtained from a path u, u u by joining u and u to a new vertex v (1 i n 1) and to a new vertex w (1 i n 1). Theorem.: DT K is difference cordial. Proof: Let V(DT K ) = V(DT ) {x : 1 i n} {v, w : 1 i n 1} and E(DT K ) = E(DT ) {u x : 1 i n} {v v, w w : 1 i n 1}. Define f: V(DT K ) {1, 6n 4} by f(u ) = 4i 1 i n f(x ) = 4i 3 1 i n f(v ) = 4i 1 1 i n 1 f(v ) = 4i 1 i n 1 f(w ) = 4n + i 3 1 i n 1 f(w ) = 4n + i 1 i n 1 Since e (1) = 4n 3 and e (0) = 4n 4, DT K is difference cordial. Theorem.3: DT K is difference cordial. Proof: Let V(DT K ) = V(DT ) {x, y : 1 i n} {v, v, w, w : 1 i n 1} and E(DT K ) = E(DT ) {u x, u y : 1 i n} {v v, v v, w w, w w : 1 i n 1}. Define f: V(DT K ) {1, 9n 6} by

3 Difference Cordial Labeling of Graphs Obtained from Double Snakes 319 f(u ) = 6i 4 1 i n f(x ) = 6i 5 1 i n f(y ) = 6i 3 1 i n f(v ) = 6i 1 1 i n 1 f(v ) = 6i 1 i n 1 f(v ) = 6i 1 i n 1 f(w ) = 6n + 3i 4 1 i n f(w ) = 6n + 3i 5 1 i n f(w ) = 6n + 3i 3 1 i n f(w ) = 6n + 3 n + + 3i 5 1 i n f w = 6n + 3 n + + 3i 4 1 i n f w = 6n + 3 n + + 3i 3 1 i n The following table (i) proves that f is a difference cordial labeling of DT K. Table (i) Values of n e (0) e (1) n 0(mod ) 11n 10 11n 8 n 1(mod ) 11n 9 11n 9 Theorem.4: DT K is difference cordial. Proof: Let V(DT K ) = V(DT ) {x, y : 1 i n} {v, v, w, w : 1 i n 1} and E(DT K ) = E(DT ) {u x, u y, x y 1 i n} {v v, v v, w w, w w : 1 i n 1}. Define f: V(DT K ) {1, 9n 6} by f(u ) = 6i 3 1 i n 1 f(x ) = 6i 4 1 i n 1 f(y ) = 6i 5 1 i n 1 f(v ) = 6i 1 i n 1 f(v ) = 6i 1 i n 1

4 30 R. Ponraj, S. Sathish Narayanan and R. Kala f(v ) = 6i 1 1 i n 1 f(w ) = 6n + 3i 5 1 i n 1 f(w ) = 6n + 3i 4 1 i n 1 f(w ) = 6n + 3i 3 1 i n 1 f(u ) = 6n 5, f(x ) = 6n 4 and f(y ) = 6n 3. Since e (1) = 7n 5 and e (0) = 7n 6, f is a difference cordial labeling of DT K. A double quadrilateral snake DQ consists of two triangular snakes that have a common path. Let V(DQ ) = {u : 1 i n} {v, w, x, y 1 i n 1} and E(DQ ) = {u u, v w, x y, w u, y u 1 i n 1}. Theorem.5: DQ K is difference cordial. Proof: Let V(DQ K ) = V(DQ ) {u : 1 i n} {v, w, x, y : 1 i n 1} and E(DQ K ) = E(DQ ) {u u : 1 i n} {v v, w w, x x, y y : 1 i n 1}. Define a map f: V(DQ K ) {1, 10n 8} by f(u ) = 4n + i 5 1 i n f(u ) = 4n + i 4 1 i n f(v ) = 4i 1 i n 1 f(v ) = 4i 3 1 i n 1 f(w ) = 4i 1 1 i n 1 f(w ) = 4i 1 i n 1 f(x ) = 6n + 4i 7 1 i n 1 f(x ) = 6n + 4i 6 1 i n 1 f(y ) = 6n + 4i 5 1 i n 1 f(y ) = 6n + 4i 4 1 i n 1 Since e (1) = 6n 5 and e (0) = 6n 6, f is a difference cordial labeling of DQ K Theorem.6: DQ K is difference cordial. Proof: Let V(DQ K ) = V(DQ ) {v, v, w, w, x, x, y, y : 1 i n 1} {u, u : 1 i n}, E(DQ K ) = E(DQ ) {v v, v v, w w, w w, x x, x x, y y : 1 i n 1} {u u, u u : 1 i n}. Define a map f: V(DQ K ) {1,, 15n 1} by f(u ) = 9i 7 1 i n f(u ) = 9i 8 1 i n f(u ) = 9i 6 1 i n f(v ) = 9i 4 1 i n 1 f(v ) = 9i 5 1 i n 1 f(v ) = 9i 3 1 i n 1

5 Difference Cordial Labeling of Graphs Obtained from Double Snakes 31 f(w ) = 9i 1 1 i n 1 f(w ) = 9i 1 i n 1 f(w ) = 9i 1 i n 1 f(x ) = 9n + 6i 10 1 i n 4 f(x ) = 9n + 6i 11 1 i n 4 f(x ) = 9n + 6i 9 1 i n 4 f x = 9n + 6 n + 6i 11 1 i 3n f x f x = 9n i 10 1 i 1 = 9n i 9 1 i 1 f(y ) = 9n + 6i 7 1 i if n 0, 1 (mod 4) 1 i if n, 3 (mod 4). f(y ) = 9n + 6i 8 1 i if n 0, 1 (mod 4) 1 i if n, 3 (mod 4). f(y ) = 9n + 6i 6 1 i if n 0, 1 (mod 4) 1 i if n, 3 (mod 4). Case (i) : n 0, 1 (mod 4). f y = 9n i 8 1 i f y 6i 7 1 i f y Case (ii) : n, 3 (mod 4). = 9n i 6 1 i. f y = 9n i 8 1 i f y 6i 7 1 i f y = 9n i 6 1 i. = 9n = 9n The following table (ii) proves that f is a difference cordial labeling of DQ K. Table (ii) Values of n e (0) e (1) n 0(mod ) 17n 16 17n 14 n 1(mod ) 17n 15 17n 15

6 3 R. Ponraj, S. Sathish Narayanan and R. Kala Theorem.7: DQ K is difference cordial. Proof: Let V(DQ K ) = V(DQ ) {v, v, w, w, x, x, y, y : 1 i n 1} {u, u : 1 i n} and E(DQ K ) {u u, u u, u u : 1 i n} {v v, v v, v v, w w, w w, w w, x x, x x, x x, y y, y y, y y : 1 i n 1}. Define a map f: V(DQ K ) {1, 15n 1} by f(u ) = 6n + 3i 8 1 i n f(u ) = 6n + 3i 7 1 i n f(u ) = 6n + 3i 6 1 i n f(v ) = 6i 3 1 i n 1 f(v ) = 6i 4 1 i n 1 f(v ) = 6i 5 1 i n 1 f(w ) = 6i 1 i n 1 f(w ) = 6i 1 i n 1 f(w ) = 6i 1 1 i n 1 f(x ) = 9n + 6i 11 1 i n 1 f(x ) = 9n + 6i 10 1 i n 1 f(x ) = 9n + 6i 9 1 i n 1 f(y ) = 9n + 6i 8 1 i n 1 f(y ) = 9n + 6i 7 1 i n 1 f(y ) = 9n + 6i 6 1 i n 1 Since e (1) = 11n 9 and e (0) = 11n 10, DQ K is a difference cordial References [1] J. A. Gallian, A Dynamic survey of graph labeling, the Electronic journal of Combinatorics 18 (011) # DS6. [] F. Harary, Graph theory, Addision wesely, New Delhi (1969). [3] R. Ponraj, S. Sathish Narayanan, R. Kala, Difference cordial labeling of graphs (Communicated). [4] R. Ponraj, S. Sathish Narayanan, R. Kala, Difference cordial labeling of corona graphs (Communicated). [5] R. Ponraj, S. Sathish Narayanan, R. Kala, Difference cordiality of some product related graphs (Communicated).