4 Remainder Cordial Labeling of Some Graphs

Transcription

1 International J.Math. Combin. Vol.(08), 8-5 Remainder Cordial Labeling of Some Graphs R.Ponraj, K.Annathurai and R.Kala. Department of Mathematics, Sri Paramakalyani College, Alwarkurichi-67, India. Department of Mathematics, Thiruvalluvar College, Papanasam -675, India. Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-670, India Abstract: Let G be a (p, q) graph. Let f be a function from V (G) to the set {,,, k} where k is an integer < k V (G). For each edge uv assign the label r where r is the remainder when f(u) is divided by f(v) (or) f(v) is divided by f(u) according as f(u) f(v) or f(v) f(u). f is called a k-remainder cordial labeling of G if v f (i) v f (j), i, j {,, k}, where v f (x) denote the number of vertices labeled with x and e f (0) e f () where e f (0) and e f () respectively denote the number of edges labeled with even integers and number of edges labelled with odd integers. A graph with admits a k-remainder cordial labeling is called a k-remainder cordial graph. In this paper we investigate the - remainder cordial behavior of grid, subdivision of crown, Subdivision of bistar, book, Jelly fish, subdivision of Jelly fish, Mongolian tent graphs. Key Words: k-remainder cordial labeling, Smarandache k-remainder cordial labeling, grid, subdivision of crown, subdivision of bistar, book, Jelly fish, subdivision of Jelly fish, Mongolian tent. AMS(00): 05C78.. Introduction We considered only finite and simple graphs.the subdivision graph S (G) of a graph G is obtained by replacing each edge uv by a path uwv. The product graph G XG is defined as follows: Consider any two points u = (u, u ) and v = (v, v ) in V = V V. Then u and v are adjacent in G G whenever [u = v and u adj v ] or [u = v and u adj v ]. The graph P m P n is called the planar grid. Let G, G respectively be (p, q ), (p, q ) graphs. The corona of G with G, G G is the graph obtained by taking one copy of G and p copies of G and joining the i th vertex of G with an edge to every vertex in the i th copy of G. A mongolian tent M m,n is a graph obtained from P m P n by adding one extra vertex above the grid and joining every other of the top row of P m P n to the new vertex. Cahit [], introduced the concept of cordial labeling of graphs. Ponraj et al. [, 6], introduced remainder cordial labeling of graphs and investigate the remainder cordial labeling behavior of path, cycle, Received February, 07, Accepted March 8, 08.

2 Remainder Cordial Labeling of Some Graphs 9 star, bistar, complete graph, S(K,n ), S(B n,n ), S(W n ), P n, P n K,n, P n Bn,n, P n B n,n, P n K,n, K,n S(K,n ), K,n S(Bn,n ), S(K,n ) S(B n,n ), etc., and also the concept of k-remainder cordial labeling introduced in [5]. In this paper we investigate the -remainder cordial labeling behavior of Grid, Subdivision of crown, Subdivision of bistar, Book, Jelly fish, Subdivision of Jelly fish, Mongolian tent, etc,. Terms are not defined here follows from Harary [] and Gallian [].. k-remainder Cordial Labeling Definition. Let G be a (p, q) graph. Let f be a function from V (G) to the set {,,, k} where k is an integer < k V (G). For each edge uv assign the label r where r is the remainder when f(u) is divided by f(v) (or) f(v) is divided by f(u) according as f(u) f(v) or f(v) f(u). The labeling f is called a k-remainder cordial labeling of G if v f (i) v f (j) and e f (0) e f (), otherwise, Smarandachely if v f (i) v f (j) or e f (0) e f () for integers i, j {,, k}, where v f (x) and e f (0), e f () respectively denote the number of vertices labeled with x, the number of edges labeled with even integers and the number of edges labelled with odd integers. Such a graph with a k-remainder cordial labeling is called a k-remainder cordial graph. First we investigate the remainder cordial labeling behavior of the planar grid. Theorem. The planar grid P m P n is remainder cordial. Proof Clearly this grid has m rows and n columns. We assign the labels to the vertices by row wise. Case. m 0 (mod ) Let m = t. Then assign the label to the vertices of st, nd,, t th rows. Next we move to the (t+) th row. Assign the label to the vertices of (t+) th, (t+) th,..., (t) th rows. Next assign the label to the vertices (t + ) th row. Assign the labels and alternatively to the vertices of (t + ) th row. Next move to (t + ) th row. Assign the labels and alternatively to the vertices of (t + ) th row. In general i th row is called as in the (i ) th row, where t + i t. This procedure continued until we reach the (t) th row. Case. m (mod ) As in Case, assign the labels to the vertices of the first, second,, (m ) th row. We give the label to the m th row as in given below. Subcase. n 0 (mod ) Rotate the row and column and result follows from Case. Subcase. n (mod ) Assign the labels,,,,,, to the vertices of the first, second,, ( ) n th columns. Next assign the label to the vertices of ( n+ )th column. Then next assign the labels,,,,,

3 0 R.Ponraj, K.Annathurai and R.Kala, to the vertices of n+, n+5,, (n )th columns. Assign the remaining vertices. Subcase. n (mod ) Assign the labels,,,,,, to the vertices of st, nd,, ( n )th columns. Next assign the label to the vertices of ( n ) th column. Then next assign the labels,,,,,, to the vertices of n +, n +,, ( n ) th columns. Finally assign the label to the remaining vertices of n th column. Subcase. n (mod ) Assign the labels,,,,,, alternatively to the vertices of st, nd,, ( n+ columns. Then next assign the labels,,,, to the vertices of n+, n+5,, ( n ) th columns. Finally assign the label to the remaining vertices of n th column. Hence f is a remainder cordial labeling of P m P n. All other cases follow by symmetry. Next is the graph K + mk. Theorem. If m 0,, (mod ) then K + mk is remainder cordial. Proof It is easy to verify that K + mk has m + vertices and m edges. Let V (K + mk ) = {u, u i, v : i m} and E(K + mk ) = {uv, uu i, vu i : i m}. Case. m 0 (mod ) Let m = t. Then assign the label, respectively to the vertices u, v. Next assign the label to the vertices u, u,, u t+. Then next assign the label to the vertices u t+, u t+,, u t+. Then followed by assign the label to the vertices u t+, u t+,...,u t. Finally assign the label to the remaining non-labelled vertices u t+, u t+,, u t. Case. m (mod ) As in Case, assign the labels to the vertices u, v, u i, ( i m ). Next assign the label to the vertex u m. Case. m (mod ) Assign the labels to the vertices u, v, u i, ( i m ) as in case(ii). Finally assign the labels, respectively to the vertices u m, u m. The table given below establish that this labeling f is a remainder cordial labeling. Nature of m e f (0) e f () m 0 (mod ) m + m m (mod ) m m + m (mod ) m m + ) th Table

4 Remainder Cordial Labeling of Some Graphs The next graph is the book graph B n. Theorem. The book B n is remainder cordial for all n. Proof Let V (B n ) = {u, v, u i, v i : i n} and E(B n ) = {uv, uu i, vv i, u i v i : i n}. Case. n is even Assign the labels, to the vertices u and v respectively. Assign the label to the vertices u, u,, u m and assign to the vertices u m +, u m +,, u n. Next we consider the vertices v i. Assign the label to the vertices v, v,, v n. Next assign the label to the remaining vertices v n +, v n +,, u n, respectively. Case. n is odd Assign the labels, to the vertices u and v respectively. Fix the labels,, to the vertices u, u,, u n + and also fix the labels,, respectively to the vertices v, v,, v n +. Assign the labels to the vertices u, u 5,, u n as in the sequence,,,...,,. In similar fashion, assign the labels to the vertices v, v 5,, v n as in the sequence,,,...,,. The table shows that this vertex labeling f is a remainder cordial labeling. Nature of n e f (0) e f () n is even m + m n is odd m m + Now we consider the subdivision of B n,n. Table Theorem.5 The subdivision of B n,n is remainder cordial. Proof Let V (S(B n,n )) = {u, v, u i, v i, w i, x, x i : i n} and E(S(B n,n )) = {uu i, vv i, u i w i, v i x i, ux, xv : i n}.it is clearly to verify that S(B n,n ) has n + vertices and n + edges. Assign the labels,, to the vertices u, x and v respectively. Assign the labels, alternatively to the vertices u, u,, u n. Next assign the labels, alternatively to the vertices w, w,, w n. We now consider the vertices v i and x i. Assign the labels, alternatively to the vertices v, v,, v n. Then finally assign the labels, alternatively to the vertices x, x,, x n. Obviously this vertex labeling is a remainder cordial labeling. Next, we consider the subdivision of crown graph. Theorem.5 The subdivision of C n K is remainder cordial. Proof Let u u... u n be a cycle C n. Let V (C n K ) = V (C n {vi : i n}) and E(C n K ) = E(C n {ui, v i : i n}).the subdivide edges u i u i+ and u i v i by x i and y i respectively. Assign the label to the vertices u i, ( i n) and to the vertices

5 R.Ponraj, K.Annathurai and R.Kala x i, ( i n). Next assign the label to the vertices y i, ( i n). Finally assign the label to the vertices v i, ( i n). Clearly, this labeling f is a remainder cordial labeling. Now we consider the Jelly fish J(m, n). Theorem.6 The Jelly fish J(m, n) is remainder cordial. Proof Let V (J(m, n)) = {u, v, x, y, u i, v j : i m and j n} and E(J(m, n)) = {uu i, vv j, ux, uy, vx, vy : i m and j n}. Clearly J(m, n) has m + n + vertices and m + n + 5 edges. Case. m = n and m is even. Assign the label to the vertices u, u,, u n and assign the label to the vertices u n +, u n +,, u n. Next assign the label to the vertices v, v,, v n and assign to the vertices v n +, v n +,..., v n. Finally assign the labels,,, respectively to the vertices u, x, y, v. Case. m = n and m is odd. In this case assign the labels to the vertices u i, v i ( i m ) and u, v, x, y as in Case. Next assign the labels, respectively to the vertices u n and v n. Case. m n and assume m > n. Assign the labels,,, to the vertices u, x, y, v respectively. As in Case and, assign the labels to the vertices u i, v i ( i n). Subcase. m n is even. Assign the labels to the vertices u n+, u n+,, u m as in the sequence,,, ;,,, ;. It is easy to verify that u n is received the label if m n 0 (mod ). Subcase. m n is odd. Assign the labels to the vertices u i (n i m) as in the sequence,,, ;,,, ;. Clearly, u n is received the label if m n 0 (mod ). For illustration, a remainder cordial labeling of Jelly fish J(m, n) is shown in Figure. Figure Theorem.8 The subdivision of the Jelly fish J(m, n) is remainder cordial.

6 Remainder Cordial Labeling of Some Graphs Proof Let V (S(J(m, n))) = {u, u i, x i, v, v j, y j : i m, j n} {w i : i 7} and E(S(J(m, n))) = {uu i, u i x i, vv j, v j y j : i m, j n} {uw, uw, w w 5, w 5 w 6, w 6 w 7, w 5 w, w v, vw, w w 7, w w 7 }. Case. m = n. Assign the label to the vertices u, u,, u m and to the vertices x, x,, x m. Next assign the label to the vertices v, v,, v m and assign the label to the vertices y, y,, y m. Finally assign the labels,,,,,,, and respectively to the vertices u, w, w 5, w 6, w 7, w, w, v and w. Case. m > n. Assign the labels to the vertices u, u i, v, v i, x i, y i, w, w, w, w, w 5, w 6, w 7, ( i n) as in case(i). Next assign the labels, to the next two vertices x n+, x n+ respectively. Then next assign the labels, respectively to the vertices x n+, x n+. Proceeding like this until we reach the vertex x n.that is the vertices x n+, x n+, x n+, x n+, are labelled in the pattern, ;, ;, ;, ;. Similarly the vertices u n+, u n+, are labelled as, ;, ;, ;,,. The Table, establish that this vertex labeling f is a remainder cordial labeling of S(J(m, n)). Nature of m and n e f (0) e f () m = n n + 5 n + 5 m > n m + n + 5 m + n + 5 Theorem.9 The graph C (t) is remainder cordial. Proof Let V (C (t) ) = {u, u i, v i : i n} and E(C (t) ) = {uu i, vv i, u i v i : i n}. Assume t. Fix the label to the central vertex u and fix the labels,,,,, respectively to the vertices u, u, u, v, v and v. Next assign the labels, to the vertices u, u 5. Then assign the labels, respectively to the next two vertices u 6, u 7 and so on. That is the vertices u, u 5, u 6, u 7 are labelled as in the pattern,,,,,. Note that the vertex u n received the label or according as n is even or odd. In a similar way assign the labels to the vertices v, v 5, v 6, v 7 as in the sequence,,,,,,. Clearly is the label of u n according as n is even or odd. The Table establish that this vertex labeling is a remainder cordial labeling of C (t), t. Nature of n e f (0) e f () n is even n n n is odd n + n + Table For t =, the remainder cordial labeling of graphs C () and C () are given below in Figure.

7 R.Ponraj, K.Annathurai and R.Kala Figure Theorem.0 The Mongolian tent M m,n is remainder cordial. Proof Assign the label to the new vertex. Case. m 0 (mod ) and n 0, (mod ). Consider the first row of M n. Assign the labels,,,,, to the vertices in the first row. Next assign the labels,,,, to the vertices in the second row. This procedure is continue until reach the n th row. Next assign the labels,,,,,, to the vertices in the n + th row. Then next assign the labels,,,,,, to the vertices in the n + th row. This proceedings like this assign the labels continue until reach the last row. Case. m (mod ) and n 0, (mod ). In this case assign the labels to the vertices as in Case. Case. m (mod ) and n 0, (mod ). Here assign the labels by column wise to the vertices of M n. Assign the labels,,,,, to the vertices in the first column. Next assign the labels,,,, to the vertices in the second column. This method is continue until reach the n th column. Next assign the labels,,,,,, to the vertices in the n + th column. Then next assign the labels,,,,,, to the vertices in the n + th column. This procedure is continue until reach the last column. Case. m (mod ) and n 0, (mod ). As in Case assign the labels to the vertices in this case. The remainder cordial labeling of graphs M 7, is given below in Figure..

8 Remainder Cordial Labeling of Some Graphs 5 Figure References [] Cahit I., Cordial Graphs : A weaker version of Graceful and Harmonious graphs, Ars Combin., (987), [] Gallian J.A., A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 9, (07). [] Harary F., Graph Theory, Addision Wesley, New Delhi, 969. [] Ponraj R., Annathurai K. and Kala R., Remainder cordial labeling of graphs, Journal of Algorithms and Computation, Vol.9, (07), 7 0. [5] Ponraj R., Annathurai K. and Kala R., k-remaider cordial graphs, Journal of Algorithms and Computation, Vol.9(), (07), -5. [6] Ponraj R., Annathurai K. and Kala R., Remaider cordiality of some graphs, Accepted for publication in Palestin Journal of Mathematics.