International J.Math. Combin. Vol.(016), 11-11 k-difference cordial labeling of graphs R.Ponraj 1, M.Maria Adaickalam and R.Kala 1.Department of Mathematics, Sri Paramakalyani College, Alwarkurichi-6741, India.Department of Economics and Statistics, Government of Tamilnadu, Chennai-600 006, India.Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-6701, India E-mail: ponrajmaths@gmail.com, mariaadaickalam@gmail.com, karthipyi91@yahoo.co.in Abstract: In this paper we introduce new graph labeling called k-difference cordial labeling. Let G be a (p,q) graph and k be an integer, k V (G). Let f : V (G) {1,,, k} be a map. For each edge uv, assign the label f(u) f(v). f is called a k- difference cordial labeling of G if v f (i) v f (j) 1 and e f (0) e f (1) 1 where v f (x) denote the number of vertices labelled with x, e f (1) and e f (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate k-difference cordial labeling behavior of star, m copies of star and we prove that every graph is a subgraph of a connected k-difference cordial graph. Also we investigate -difference cordial labeling behavior of some graphs. Key Words: Path, complete graph, complete bipartite graph, star, k-difference cordial labeling, Smarandachely k-difference cordial labeling. AMS(010): 05C78. 1. Introduction All graphs in this paper are finite and simple. The graph labeling is applied in several areas of sciences and few of them are coding theory, astronomy, circuit design etc. For more details refer Gallian []. Let G 1, G respectively be (p 1, q 1 ), (p, q ) graphs. The corona of G 1 with G, G 1 G is the graph obtained by taking one copy of G 1 and p 1 copies of G and joining the i th vertex of G 1 with an edge to every vertex in the i th copy of G. The subdivision graph S (G) of a graph G is obtained by replacing each edge uv by a path uwv. The union of two graphs G 1 and G is the graph G 1 G with V (G 1 G ) = V (G 1 ) V (G ) and E (G 1 G ) = E (G 1 ) E (G ). In [1], Cahit introduced the concept of cordial labeling of graphs. Recently Ponraj et al. [4], introduced difference cordial labeling of graphs. In this way we introduce k-difference cordial labeling of graphs. Also in this paper we investigate the k-difference cordial labeling behavior of star, m copies of star etc. x denote the smallest integer less than or equal to x. Terms and results not here follows from Harary []. 1 Received June 4, 015, Accepted May 6, 016.
1 R.Ponraj, M.Maria Adaickalam and R.Kala. k-difference Cordial Labeling Definition.1 Let G be a (p, q) graph and k be an integer k V (G). Let f : V (G) {1,,, k} be a function. For each edge uv, assign the label f(u) f(v). f is called a k-difference cordial labeling of G if v f (i) v f (j) 1 and e f (0) e f (1) 1, and Smarandachely k-difference cordial labeling if v f (i) v f (j) > 1 or e f (0) e f (1) > 1, where v f (x) denote the number of vertices labelled with x, e f (1) and e f (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling or Smarandachely k-difference cordial labeling is called a k-difference cordial graph or Smarandachely k-difference cordial graph, respectively. Remark. (1) p-difference cordial labeling is simply a difference cordial labeling; () -difference cordial labeling is a cordial labeling. Theorem. Every graph is a subgraph of a connected k-difference cordial graph. Proof Let G be (p, q) graph. Take k copies of graph K p. Let G i be the i th copy of K p. Take k copies of the K ( p ) and the ith copies of the K ( p ) is denoted by G i. Let V (G i) = {u j i : 1 j k, 1 i p}. Let V (G i ) = {vj i : 1 j k, 1 i p}. The vertex and edge set of super graph G of G is as follows: Let V (G ) = k V (G i ) k V (G i ) {w i : 1 i k} {w}. E(G ) = k i=1 i=1 i=1 E(G i ) {u j 1 vj i : 1 i ( ) p, 1 j k 1} {u k 1 w, wvi k : 1 i ( p ) } {u j p w j : 1 j k} {u j uj+1 : 1 j k 1} {w 1 w }. Assign the label i to the vertices of G i, 1 i k. Then assign the label i + 1 to the vertices of G i, 1 i k 1. Assign the label 1 to the vertices of G k. Then assign to the vertex w. Finally assign the label i to the vertex w i, 1 i k. Clearly v f (i) = p + ( p ) + 1, i = 1,,...,k, v f () = p + ( ) p + and ef (1) = k ( ) p + k, ef (0) = k ( p ) + k + 1. Therefore G is a k-difference cordial graph. Theorem.4 If k is even, then k-copies of star K 1,p is k-difference cordial. Proof Let G i be the i th copy of the star K 1,p. Let V (G i ) = {u j, v j i : 1 j k, 1 i p} and E(G i ) = {u j v j i : 1 j k, 1 i p}. Assign the label i to the vertex u j, 1 j k. Assign the label i + 1 to the pendent vertices of G i, 1 i k. Assign the label k i + 1 to the pendent vertices of G k +i, 1 i k 1. Finally assign the label 1 to all the pendent vertices of the star G k. Clearly, v f (i) = p + 1, 1 i k, e f (0) = e f (1) = kp. Therefore f is a k-difference cordial labeling of k-copies of the star K 1,p. Theorem.5 If n 0 (mod k) and k 6, then the star K 1,n is not k-difference cordial. Proof Let n = kt. Suppose f is a k-difference cordial labeling of K 1,n. Without loss of generality, we assume that the label of central vertex is r, 1 r k. Clearly v f (i) = t,
k-difference cordial labeling of graphs 1 1 i n and i r, v f (r) = t + 1. Then e f (1) t and e f (0) (k )t. Now e f (0) (k )t t (k 4)t, which is a contradiction. Thus f is not a k-difference cordial. Next we investigate -difference cordial behavior of some graph.. -Difference Cordial Graphs First we investigate the path. Theorem.1 Any path is -difference cordial. Proof Let u 1 u...u n be the path P n. The proof is divided into cases following. Case 1. n 0 (mod 6). Let n = 6t. Assign the label 1,,, 1,, to the first consecutive 6 vertices of the path P n. Then assign the label,, 1,,, 1 to the next 6 consecutive vertices. Then assign the label 1,,, 1,, to the next six vertices and assign the label,, 1,,, 1 to the next six vertices. Then continue this process until we reach the vertex u n. Case. n 1 (mod 6). This implies n 1 0 (mod 6). Assign the label to the vertices of u i, 1 i n 1 as in case 1. If u n 1 receive the label, then assign the label to the vertex u n ; if u n 1 receive the label 1, then assign the label 1 to the vertex u n. Case. n (mod 6). Therefore n 1 1 (mod 6). As in case, assign the label to the vertices u i, 1 i n 1. Next assign the label to u n. Case 4. n (mod 6). This forces n 1 (mod 6). Assign the label to the vertices u 1, u,... u n 1 as in case. Assign the label 1 or to u n according as the vertex u n receive the label or 1. Case 5. n 4 (mod 6). This implies n 1 (mod 6). As in case 4, assign the label to the vertices u 1, u,, u n 1. Assign the label or 1 to the vertex u n according as the vertex u n 1 receive the label 1 to. Case 6. n 5 (mod 6). This implies n 1 4 (mod 6). Assign the label to the vertices u 1, u,, u n 1 as in Case 5. Next assign the label to u n. Example. A -difference cordial labeling of the path P 9 is given in Figure 1. 1 1 1 Figure 1
14 R.Ponraj, M.Maria Adaickalam and R.Kala Corollary. If n 0, (mod 4), then the cycle C n is -difference cordial. Proof The vertex labeling of the path given in Theorem.1 is also a -difference cordial labeling of the cycle C n. Theorem.4 The star K 1,n is -difference cordial iff n {1,,, 4, 5, 6, 7, 9}. Proof Let V (K 1,n ) = {u, u i : 1 i n} and E(K 1,n ) = {uu i : 1 i n}. Our proof is divided into cases following. Case 1. n {1,,, 4, 5, 6, 7, 9}. Assign the label 1 to u. The label of u i is given in Table 1. n\u i u 1 u u u 4 u 5 u 6 u 7 u 8 u 9 1 1 Case. n / {1,,, 4, 5, 6, 7, 9}. 4 1 5 1 6 1 7 1 9 1 1 Table 1 Let f(u) = x where x {1,, }. To get the edge label 1, the pendent vertices receive the label either x 1 or x + 1. Subcase 1. n = t. Subcase 1a. x = 1 or x =. When x = 1, e f (1) = t or t + 1 according as the pendent vertices receives t s or (t+1) s. Therefore e f (0) = t or t 1. Thus e f (0) e f (1) = t > 1, t > 4 a contradiction. When x =, e f (1) = t or t + 1 according as the pendent vertices receives t s or (t+1) s. Therefore e f (0) = t or t 1. Thus e f (0) e f (1) = t or t. Therefore, e f (0) e f (1) > 1, a contradiction. Subcase 1b. x =. In this case, e f (1) = t or t + 1 according as pendent vertices receives t s or (t-) s. Therefore e f (0) = t or t 1. e f (1) e f (0) = t or t + as t >. Therefore, e f (0) e f (1) > 1, a contradiction. Subcase. n = t + 1.
k-difference cordial labeling of graphs 15 Subcase a. x = 1 or. Then e f (1) = t or t +1 according as pendent vertices receives t s or (t+1) s. Therefore e f (0) = t + 1 or t. e f (0) e f (1) = t + 1 or t 1 as t >. Therefore, e f (0) e f (1) >, a contradiction. Subcase b. x =. In this case e f (1) = t or t + 1 according as pendent vertices receives t s 1 and t s and t s 1 and (t+) s. Therefore e f (0) = t + 1 or t. e f (1) e f (0) = t 1 or t as t >. Therefore, e f (0) e f (1) > 1, a contradiction. Subcase. n = t +. Subcase a. x = 1 or. This implies e f (1) = t + 1 and e f (0) = t + 1. e f (0) e f (1) = t as t >. Therefore, e f (0) e f (1) > 1, a contradiction. Subcase b. x =. This implies e f (1) = t + and e f (0) = t. e f (1) e f (0) = t + as t > 1. Therefore, e f (1) e f (0) > 1, a contradiction. Thus K 1,n is -difference cordial iff n {1,,, 4, 5, 6, 7, 9}. Next, we research the complete graph. Theorem.5 The complete graph K n is -difference cordial if and only if n {1,,, 4, 6, 7, 9, 10}. Proof Let u i, 1 i n be the vertices of K n. The -difference cordial labeling of K n, n {1,,, 4, 6, 7, 9, 10} is given in Table. n\u i u 1 u u u 4 u 5 u 6 u 7 u 8 u 9 u 10 1 1 1 1 4 1 1 6 1 1 7 1 1 1 9 1 1 1 10 1 1 1 Table Assume n / {1,,, 4, 6, 7, 9, 10}. Suppose f is a -difference cordial labeling of K n. Case 1. n 0 (mod ). Let n = t, t >. Then v f (0) = v f (1) = v f () = t. This implies e f (0) = ( t ) + ( t ) + ( t ) +
16 R.Ponraj, M.Maria Adaickalam and R.Kala t = 5t t. Therefore e f (1) = t + t = t. e f (0) e f (1) = 5t t t > 1 as t >, a contradiction. Case. n 1 (mod ). Let n = t + 1, t >. Subcase 1. v f (1) = t + 1. Therefore v f () = v f () = t. This forces e f (0) = ( ) ( t+1 + t ) ( + t ) + t(t + 1) = 1 (5t + t). e f (1) = t(t + 1) + t = t + t. Then e f (0) e f (1) = 1 (5t + t) (t + t) > 1 as t >, a contradiction. Subcase. v f () = t + 1. Similar to Subcase 1. Subcase. v f () = t + 1. Therefore v f (1) = v f () = t. In this case e f (0) = 5t +t and e f (1) = t(t + 1) + t(t + 1) = t + t. This implies e f (0) e f (1) = 5t +t (t + t) > 1 as t >, a contradiction. Case. n (mod ). Let n = t +, t 1. Subcase 1. v f (1) = t. Therefore v f () = v f () = t + 1. This gives e f (0) = ( t ) + ( t+1 ) + ( t+1 ) + t(t + 1) = 5t +t and e f (1) = t(t+1)+(t+1) = t +t+1. This implies e f (0) e f (1) = 5t +t (t +t+1) > 1 as t 1, a contradiction. Subcase. v f () = t. Similar to Subcase 1. Subcase. v f () = t. Therefore v f (1) = v f () = t + 1. In this case e f (0) = ( ) ( t+1 + t+1 ) ( + t ) + (t + 1)(t + 1) = 5t +5t+ and e f (1) = t(t + 1) + t(t + 1) = t + t. This implies e f (0) e f (1) = 5t +5t+ (t + t) > 1 as t 1, a contradiction. Theorem.6 If m is even, the complete bipartite graph K m,n (m n) is -difference cordial. Proof Let V (K m,n ) = {u i, v j : 1 i m, 1 j n} and E(K m,n ) = {u i v j : 1 i
k-difference cordial labeling of graphs 17 m, 1 j n}. Define a map f : V (K m,n ) {1,, } by f(u i ) = 1, 1 i m f(u m +i ) =, 1 i m f(v i ) =, 1 i f(v +i ) = 1, 1 i m 1 if m + n 1, (mod ) 1 i m if m + n 0 (mod ) f(v m 1+i) =, 1 i n + m + 1 if m + n 1, (mod ) f(v m+i) =, 1 i n + m if m + n 0 (mod ) Since e f (0) = e f (1) = mn, f is a -difference cordial labeling of K m,n. Example.7 A -difference cordial labeling of K 5,8 is given in Figure. 1 1 1 1 Next, we research some corona of graphs. Figure Theorem.8 The comb P n K 1 is -difference cordial. Proof Let P n be the path u 1 u... u n. Let V (P n K 1 ) = V (P n ) {v i : 1 i n} and E(P n K 1 ) = E(P n ) {u i v i : 1 i n}. Case 1. n 0 (mod 6). Define a map f : V (G) {1,, } by f(u 6i 5 ) = f(u 6i ) = 1, 1 i n 6 f(u 6i 4 ) = f(u 6i 1 ) =, 1 i n 6 f(u 6i ) = f(u 6i ) =, 1 i n 6. In this case, e f (0) = n 1 and e f (1) = n. Case. n 1 (mod 6). Assign the label to the vertices u i, v i (1 i n 1) as in case 1. Then assign the labels 1, to the vertices u n, v n respectively. In this case, e f (0) = n 1, e f (1) = n.
18 R.Ponraj, M.Maria Adaickalam and R.Kala Case. n (mod 6). As in Case, assign the label to the vertices u i, v i (1 i n 1). Then assign the labels, to the vertices u n, v n respectively. In this case, e f (0) = n, e f (1) = n 1. Case 4. n (mod 6). Assign the label to the vertices u i, v i (1 i n 1) as in case. Then assign the labels, 1 to the vertices u n, v n respectively. In this case, e f (0) = n 1, e f (1) = n. Case 5. n 4 (mod 6). As in Case 4, assign the label to the vertices u i, v i (1 i n 1). Then assign the labels, to the vertices u n, v n respectively. In this case, e f (0) = n 1, e f (1) = n. Case 6. n 5 (mod 6). Assign the label to the vertices u i, v i (1 i n 1) as in case 5. Then assign the labels, 1 to the vertices u n, v n respectively. In this case, e f (0) = n 1, e f (1) = n. Therefore P n K 1 is -difference cordial. Theorem.9 P n K 1 is -difference cordial. Proof Let P n be the path u 1 u u n. Let V (P n K 1 ) = V (P n ) {v i, w i : 1 i n} and E(P n K 1 ) = E(P n ) {u i v i, u i w i : 1 i n}. Case 1. n is even. Define a map f : V (P n K 1 ) {1,, } as follows: f(u i 1 ) = 1, 1 i n f(u i ) =, 1 i n f(v i 1 ) = 1, 1 i n f(v i ) =, 1 i n f(w i ) =, 1 i n. In this case, v f (1) = v f () = v f () = n, e f (0) = n and e f(1) = n 1. Case. n is odd. Define a map f : V (P n K 1 ) {1,, } by f(u 1 ) = 1, f(u ) =, f(u ) =, f(v 1 ) = f(v ) = 1, f(w 1 ) = f(w ) =, f(v ) = f(w ) =, f(u i+ ) =, 1 i n f(u i+ ) = 1, 1 i n f(v i+ ) =, 1 i n f(v i+ ) = 1, 1 i n f(w i+ ) =, 1 i n.
k-difference cordial labeling of graphs 19 Clearly, v f (1) = v f () = v f () = n, e f (0) = e f (1) = n 1. Next we research on quadrilateral snakes. Theorem.10 The quadrilateral snakes Q n is -difference cordial. Proof Let P n be the path u 1 u u n. Let V (Q n ) = V (P n ) {v i, w i : 1 i n 1} and E(Q n ) = E(P n ) {u i v i, v i w i, w i u i+1 : 1 i n 1}. Note that V (Q n ) = n and E(Q n ) = 4n 4. Assign the label 1 to the path vertices u i, 1 i n. Then assign the labels, to the vertices v i, w i 1 i n 1 respectively. Since v f (1) = n, v f () = v f () = n 1, e f (0) = e f (1) = n, f is a -difference cordial labeling. The next investigation is about graphs B n,n, S(K 1,n ), S(B n,n ). Theorem.11 The bistar B n,n is -difference cordial. Proof Let V (B n,n ) = {u, v, u i, v i : 1 i n} and E(B n,n ) = {uv, uu i, vv i : 1 i n}. Clearly B n,n has n + vertices and n + 1 edges. Case 1. n 0 (mod ). Assign the label 1, to the vertices u and v respectively. Then assign the label 1 to the vertices u i, v i (1 i n ). Assign the label to the vertices u n +i, v n +i (1 i n ). Finally assign the label to the vertices u n +i, vn +i (1 i n ). In this case e f(1) = n + 1 and e f (0) = n. Case. n 1 (mod ). Assign the labels to the vertices u, v, u i, v i (1 i n 1) as in Case 1. Then assign the label, to the vertices u n, v n respectively. In this case e f (1) = n and e f (0) = n + 1. Case. n (mod ). As in Case, assign the label to the vertices u, v, u i, v i (1 i n 1). Finally assign 1, to the vertices u n, v n respectively. In this case e f (1) = n and e f (0) = n + 1. Hence the star B n,n is -difference cordial. Theorem.1 The graph S(K 1,n ) is -difference cordial. Proof Let V (S(K 1,n )) = {u, u i, v i : 1 i n} and E(S(K 1,n )) = {uu i, u i v i : 1 i n}. Clearly S(K 1,n ) has n + 1 vertices and n edges. Case 1. n 0 (mod ). Define a map f : V (S(K 1,n )) {1,, } as follows: f(u) =, f(u i ) f(u t+i ) f(v i ) = 1, 1 i t =, 1 i t =, 1 i t f(v t+i ) = 1, 1 i t.
10 R.Ponraj, M.Maria Adaickalam and R.Kala Case. n 1 (mod ). As in Case 1, assign the label to the vertices u, u i, v i (1 i n 1). Then assign the label 1, to the vertices u n, v n respectively. Case. n (mod ). As in Case, assign the label to the vertices u, u i, v i (1 i n 1). Then assign the label, 1 to the vertices u n, v n respectively. f is a -difference cordial labeling follows from the following Table. Values of n v f (1) v f () v f () e f (0) e f (1) n = t t t + 1 t t t n = t + 1 t + 1 t + 1 t + 1 t + 1 t + 1 n = t + t + t + t + 1 t + t + Table Theorem.1 S(B n,n ) is -difference cordial. Proof Let V (S(B n,n )) = {u, w, v, u i, w i, v i, z i : 1 i n} and E(S(B n,n )) = {uw, wv, uu i, u i w i, vv i, v i z i : 1 i n}. Clearly S(B n,n ) has 4n + vertices and 4n + edges. Case 1. n 0 (mod ). Define a map f : V (S(B n,n )) {1,, } by f(u) = 1, f(w) =, f(v) =, f(w i ) = f(v i ) = f(z i ) =, 1 i n 1, 1 i n, 1 i n f(u i ) = 1, 1 i n f(u n +i ) =, 1 i n f(u n +i ) =, 1 i n. Case. n 1 (mod ). As in Case 1, assign the label to the vertices u, w, v, u i, v i, w i, z i (1 i n 1). Then assign the label 1,, 1, to the vertices u n, w n, v n, z n respectively. Case. n (mod ). As in Case, assign the label to the vertices u, w, v, u i, v i, w i, z i (1 i n 1). Then assign the label,, 1, to the vertices u n, w n, v n, z n respectively. f is a -difference cordial labeling follows from the following Table 4.
k-difference cordial labeling of graphs 11 Values of n v f (1) v f () v f () e f (0) e f (1) n 0 (mod ) 4n+ n 1 (mod ) 4n+5 n (mod ) 4n+4 4n+ 4n+ 4n+4 4n+ 4n+ 4n+1 4n+ 4n+ 4n+ 4n+ 4n+ 4n+ Table 4 Finally we investigate cycles C (t) 4. Theorem.14 C (t) 4 is -difference cordial. Proof Let u be the vertices of C (t) 4 and i th cycle of C (t) 4 be uu i 1 ui ui u. Define a map f from the vertex set of C (t) 4 to the set {1,, } by f(u) = 1, f(u i ) =, 1 i t, f(u i 1) = 1, 1 i t, f(u i ) =, 1 i t. Clearly v f(1) = t + 1, v f () = v f () = t and e f (0) = e f (1) = t. Hence f is -difference cordial. References [1] I.Cahit, Cordial Graphs: A weaker version of Graceful and Harmonious graphs, Ars Combin., (1987), 01-07. [] J.A.Gallian, A Dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 17 (014) #Ds6. [] F.Harary, Graph Theory, Addision wesley, New Delhi (1969). [4] R.Ponraj, S.Sathish Narayanan and R.Kala, Difference cordial labeling of graphs, Global Journal of Mathematical Sciences: Theory and Practical, 5(01), 185-196.