LOCAL CONNECTIVE CHROMATIC NUMBER OF DIRECT PRODUCT OF PATHS AND CYCLES

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1 BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) , ISSN (o) /JOURNALS / BULLETIN Vol. 7(017), DOI: /BIMVI Ç Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN (o), ISSN X (p) LOCAL CONNECTIVE CHROMATIC NUMBER OF DIRECT PRODUCT OF PATHS AND CYCLES Canan Çiftçi and Pinar Dündar Abstract. Graph coloring is one of the most important concept in graph theory. There are many types of coloring. We study on the local connective chromatic number of a graph G that is defined by us. In this paper, we determine the local connective chromatic number of the direct product of two paths P m P n, two cycles C m C n and for the direct product of a cycle and a path C m P n, where m and n are the number of vertices. 1. Introduction Let G be a simple undirected graph, where V (G) and E(G) denote the set of vertices and the set of edges of G, respectively. For two vertices u, v V (G), u and v are adjacent if they are joined by an edge. Two vertices that are not adjacent in a graph G are said to be independent. The independence number β(g) of a graph G is the maximum cardinality among the independent sets of vertices of G. For the notations and terminology not defined here, we follow [6]. The connectivity κ = κ(g) of a graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph. Two paths are internally disjoint (vertex disjoint) if they do not share a common vertex except their end vertices. The local connectivity κ G (u, v) = κ(u, v) between two distinct vertices u and v of a graph G is defined as the smallest number of vertices whose removal separates u and v. By Menger s theorem [14], κ(u, v) equals the maximum number of internally disjoint u v paths in G and κ(g) = min{κ(u, v) : u, v V (G)}. It is straightforward to verify that κ(g) δ(g) and κ(u, v) min{deg(u), deg(v)} [16]. 010 Mathematics Subject Classification. 05C15, 05C76. Key words and phrases. Local connective chromatic number, Internally disjoint path, Direct product. 561

2 56 C. ÇIFTÇI AND P. DÜNDAR The local connective coloring is defined by us by inspiring the notion of packing coloring [5, 9, 1, 18]. Routing is the process of delivering messages among vertices and selecting the best paths in a network. Efficiency and reliability of routing can be achieved by using internally disjoint paths because the failure of a path would not affect the performance of other paths. Then the more internally disjoint paths are the better for a network [13]. Thus, we use the term internally disjoint path in our coloring and color the vertices depending on the number of internally disjoint paths between two vertices. A graph G which has a local connective k- coloring can be partitioned into disjoint color classes X 1, X,..., X k and can be drawn as a k-partite graph. Thereby, the graph is partitioned into the subsets which have disjoint paths. Looking for a secure disjoint path between two vertices u and v in any color class X i, we make this search with the vertices in the other color classes. This indicate that we look for disjoint paths starting from u and ending to v using the vertices in the other color classes. Thus, this search can be made with V (G) ( X i ) vertices. Thereby, NP-complete problem can be solved more easily. Local connective coloring provides to facilitate the routing of non adjacent vertices to communicate with each other. The direct product G H of two graphs G and H is a graph with V (G H) = V (G) V (H) and E(G H) = {(u 1, v 1 )(u, v ) : u 1 u E(G) and v 1 v E(H)}. It is also known as Kronecker product, tensor product, categorical product and graph conjunction. This graph product is commutative and associative [3]. The direct product of graphs has been extensively investigated concerning graph recognition and decomposition, graph embeddings, matching theory and stability in graphs [1, 4]. More generally, the direct product is a widely used tool in the area of graph colorings [11]. Lemma 1.1. [17] Let G be a connected graph. If G has no odd cycle, then G K has exactly two connected components isomorphic to G. Theorem 1.1. [17] Let G and H be connected graphs. The graph G H is connected if and only if any G or H contains an odd cycle. Corollary 1.1. [17] If G and H are connected graphs with no odd cycles then G H has exactly two connected components. Theorem 1.. [15] Let G = (V, E) be a connected graph, and H = (V 1, V, E ) be a bipartite connected graph, then G H is a bipartite graph, the partition of the vertex set is (V V 1 ) and (V V ). Theorem 1.3. [8] The direct product of two connected graphs is a non connected graph if and only if both are bipartite. Lemma 1.. [10] If G = (V 0 V 1, E) and H = (W 0 W 1, F ) are bipartite graphs, then (V 0 W 0 ) (V 1 W 1 ) and (V 0 W 1 ) (V 1 W 0 ) are vertex sets of the two components of G H. Lemma 1.3. [10] If G is a connected, bipartite graph and n 4 is an even integer, then the graph G C n consists of two isomorphic connected components.

3 LOCAL CONNECTIVE CHROMATIC NUMBER OF DIRECT PRODUCT OF Theorem 1.4. [] If G and H are regular graphs then G H is also a regular graph.. Local Connective Chromatic Number of Direct Product Graphs Definition.1. A local connective k-coloring of a graph G is a mapping c : V (G) {1,,..., k} such that (1) If uv E(G), then c(u) c(v), and () If uv / E(G) and c(u) = c(v) = i, then κ(u, v) i, where κ(u, v) is the maximum number of internally disjoint paths between u and v. The smallest integer k for which there exists a local connective k- coloring of G is called the local connective chromatic number of G, and it is denoted by χ lc (G). The first condition characterizes proper coloring. Thus, every local connective coloring is a proper coloring. The vertices of G are partitioned into disjoint color classes X 1, X,..., X k, where each color class X i consists of distinct vertices u, v X i such that κ(u, v) i and n X i = V (G). The maximum cardinality of X i in G is denoted by k i. i=1 In this section, we give local connective chromatic number of direct product of paths and cycles. Let G and H be any two graphs with vertex sets V (G) = {u 1, u,..., u m }, V (H) = {v 1, v,..., v n }, respectively. A vertex (u i, v j ) is abbreviated as w ij, where w ij V (G H), i {1,,..., m}, j {1,,..., n}. Theorem.1. Let P m and P n be two paths of order m and n, respectively. Then, 4 χ lc (P m P n ) max{n + 5, m + n + 7} for m n. Proof. It is known that paths are bipartite graphs. By Theorem 1.3, the graph P m P n is non connected for m and n being odd or even. Further by Corollary 1.1, P m P n has two connected components as G 1 and G. Since δ(g 1 ) = δ(g ) = and (G 1 ) = (G ) = 4, we have κ(w ij, w kl ) 4, where w ij, w kl V (G 1 )(or V (G )), i, k {1,,..., m}, j, l {1,,..., n}. Thus, k i 1 for i 5. That is, the pair of vertices can be colored with the same color at most color 4, and the remaining uncolored vertices receive different colors. We prove this theorem in four cases for m and n being odd or even. Case 1. Let m and n be odd. Since V (P m P n ) = mn, we have V (G 1 ) = mn, V (G ) = mn. The graph P m P n has four vertices of each of degree one in the only one component. Assume that these vertices be in the component G 1. Since there is one internally disjoint path between these vertices, they can be colored with color 1. The vertex w ij can be colored with color 1, where i {1, 3,..., m}, j {1, 3,..., n}. Thus, β(g 1 ) = n m vertices in G 1 are colored with color 1, and n m vertices in G 1 remain uncolored. n m < mn and thus color the graph G with color. When we start coloring from the vertex w 1 and color all vertices which are not adjacent with each other in G, then maximum n m vertices in G are colored

4 564 C. ÇIFTÇI AND P. DÜNDAR Figure 1. Local connective coloring of P 5 P 7 with color. Thus, there are n m and n m uncolored vertices in G and G, respectively. Case 1.1. Let m = 3 and n 3. For all vertices w ij, w kl in G 1 or G, we have κ(w ij, w kl ), where i, k {1,, 3}, j, l {1,,..., n}. Thus, there is not any vertex colored with color 3 and color 4. Hence, the remaining n m vertices are colored with different colors. Then we have χ lc (P 3 P n ) = + n m = n +. Case 1.. Let m = 5 and n 5. Since κ(g 1 ) = 1, κ(g ) =, we have κ(w ij, w kl ), where w ij, w kl V (G ). Four vertices in P 5 P n are colored with color 3. Color the vertex w ij in G with color 3 for the minimum local connective coloring number, where i {, 4}, j {3, 5}. Case If m = n = 5, no vertices that remain uncolored in G 1 or G are colored with color 4. Thus, the remaining vertices receive different colors and we have χ lc (P 5 P 5 ) = 3 + n m 4 = 9. Case 1... If m = 5 and n 7, the vertex w i4 for i {, 4} in G 1 is colored with color 4. The remaining n m 6 = n 6 vertices receive different colors. Thus, we have χ lc (P 5 P n ) = n. Case 1.3. Let m 7 and n 7. Take the vertex w ij in G 1, where i {, 4,..., m 1}, j {, 4,..., n 1}. The number of internally disjoint paths between these vertices is at most 4. Thus, maximum ( m 1 n 1 )( ) vertices are colored with color 3, and there is no vertex in G 1 remains uncolored.

5 LOCAL CONNECTIVE CHROMATIC NUMBER OF DIRECT PRODUCT OF Case If m = n = 7, the number of internally disjoint paths between only two vertices in G is 4. Hence, these vertices receive color 4, and we have χ lc (P 7 P 7 ) = 4 + m n = 14. Case Let m = 7 and n 9. Since m = 7, there are four vertices in G that the number of internally disjoint paths between them is 4. Thus, these four vertices receive color 4, and we have χ lc (P 7 P n ) = 4 + m n 4 = 3 (n + 1). Case Let m 9 and n 9. Take the vertices w ij, where i {4, 6, 8,..., m 3}, j {3, 5, 7,..., n } and w kl, where k {, m 1}, l {5, 7, 9,..., n 4} in G. Since the number of internally disjoint paths between them is 4, m 4 n 3 n 6 ( ) + vertices are colored with color 4. Hence, the remaining m + 3 vertices receive m + 3 different colors, and we have χ lc (P m P n ) = 4 + m + 3 = m + 7. Consequently, if m and n are odd, then we have n +, m = 3, n 3 9, m = n = 5 n, m = 5, n 7 χ lc (P m P n ) = 14, m = n = 7 3 (n + 1), m = 7, n 9 m + 7, m 9, n 9. Case. Let m be even and n be odd. V (P m P n ) = mn and V (G 1 ) = V (G ) = mn. The graph P m P n has four vertices of each of degree one. Two of them are in G 1 and the other two vertices are in G. Assume that we start coloring from the vertex w 11 in G 1. Assign color 1 to every vertex when i and j are odd. Thus, β(g 1 ) = m n vertices are colored with color 1. The graph G 1 has n uncolored vertices of degree and n ( m 1) uncolored vertices each of degree 4. Then the total number of the remaining uncolored vertices in G 1 is n m. Start coloring the graph G with color for the minimum number of local connective coloring. Assume that we start coloring from the vertex w 1. The graph mn G has total vertices of degree two and four, and since the number of internally disjoint paths between these vertices is at least, each vertex which is not adjacent with each other is colored with color. Thus, the vertices w ij, where i {, 4, 6,..., m }, j {1, 3, 5,..., n} and w nl, where l {3, 5, 7,..., n } are colored with color. Then maximum n m ( ) + n 3 = m n vertices in G receive color. Among the remaining + n m vertices in G, two of them have degree one, n of them are each of degree and n ( m 1) of them are each of degree 4. Thus, there are + n m vertices in the graph P m P n remain uncolored. Case.1. Let m = and n 3. In this case, P n = G1, P n = G and P P n = Pn. Since χ lc (P n ) = 1 + n

6 566 C. ÇIFTÇI AND P. DÜNDAR by [7], we get χ lc (P P n ) = n + χ lc (P n ) = n n = 3n + 1. Case.. Let m = 4 and n 5. Since κ(w ij, w kl ), where w ij, w kl V (G 1 )(or V (G )) there is not any vertex colored with color 3 and 4. Hence, the remaining m n + = n vertices receive different colors. Then we haveχ lc (P 4 P n ) = n +. Case.3. Let m 6 and n 7. Every pair of vertices in G 1 each of degree 4 satisfy the condition κ(w ij, w kl ) 3, where i, k {, 4,..., m }, j, l {, 4,..., n 1}. Thus, n ( m 1) vertices are colored with color 3. The number of the remaining vertices in G 1 is n and each of them has degree two. Further, these vertices satisfy the condition κ(w mj, w ml ), where j, l {, 4,..., n 1},j l. Then they receive different colors. Hence, we have χ lc (G 1 ) = + n. Consider coloring the graph G. For the vertices w ij, where i {5, 7, 9,..., m 3}, j {, 4, 6,..., n 1} and w kl, where k {3, m 1}, l {4, 6, 8,..., n 3}, the number of internally disjoint paths between them are 4 and so m 5 n 1 n 4 ( )+ vertices are colored with color 4. The remaining n+11 vertices in G receive different colors. Thus, χ lc (G ) = + n+11, and we have χ lc (P m P n ) = χ lc (G 1 ) + χ lc (G ) = n + 9. Consequently, if m is even and n is odd, then we have 3n+1, m =, n 3 χ lc (P m P n ) = n +, m = 4, n 5 n + 9, m 6, n 7. Case 3. Let m be odd and n be even. The graph P m P n has four vertices of each of degree one. Two of them are in G 1 and the other two vertices are in G. If we color the vertices of G 1 starting from the vertex w 11 as Case, n m vertices receive color 1. For color, assume that we start coloring from the vertex w 1 in G. The vertices w ij, where i {3, 5, 7,..., m }, j {, 4, 6,..., n} and w kl, where k {1, m}, l {, 4, 6,..., n } are colored with color. There are at most (m 3)n 4 + (n ) = m n vertices which receive color. Thus, m n + vertices remain uncolored in P m P n. Case 3.1. Let m = 3 and n 4. In this case, κ(w ij, w kl ), where w ij, w kl V (G 1 )(or V (G )), i, k {1,, 3}, j, l {1,,..., n}. Thus, there is not any pair of vertices that is colored with color 3 or color 4. Then we have χ lc (P 3 P n ) = + m n + = n + 4. Case 3.. Let m = 5 and n 6. If κ(w ij, w kl ) 3 or deg(w ij ) 3, where i, k {1,,..., m}, j, l {1,,..., n}, any two vertices w ij and w kl can be colored with color 3. Thus, there are m ( n 1)

7 LOCAL CONNECTIVE CHROMATIC NUMBER OF DIRECT PRODUCT OF vertices in G 1 each of degree 4 that can be colored with color 3. Since m = 5, only four vertices of them are colored with color 3. Case If m = 5 and n = 6, there is not any pair of vertices which receives color 4. Hence, the remaining vertices are colored with different colors, and so we have χ lc (P 5 P 6 ) = 3 + m n + 4 = 13. Case 3... Let m = 5 and n 8. The number of internally disjoint paths between only two vertices in G is 4. Thus, these two vertices are colored with color 4. Since the remaining n 4 vertices receive different colors, we have χ lc (P 5 P n ) = n. Case 3.3. Let m 7 and n 8. Since m 7, in G 1 there are m ( n 1) vertices each of degree 4 which are colored with color 3. Thus, we have χ lc (G 1 ) = + m n m (n 1) = + m. Case If m = 7 and n 8, only four vertices in G can be colored with color 4. Since the remaining n m vertices in G receive different colors, χ lc (G ) = + n m = 3n, and we have χ lc(p 7 P n ) = χ lc (G 1 ) + χ lc (G ) = 3n + 5. Case Let m 9 and n 10. For the minimum number of local connective coloring consider the vertices w ij, where i {4, 6,..., m 3}, j {3, 5,..., n 1} and w kl, where k {, m 1}, l {5, 7,..., n 3} in G. Since the number of internally disjoint paths between these m 4 n n 5 ( ) + vertices is 4, they are colored with color 4. The remaining m+11 vertices in G receive different colors. Hence, χ lc (G ) = + m+11 and we have χ lc (P m P n ) = χ lc (G 1 ) + χ lc (G ) = m + 9. Consequently, if m is odd and n is even, then we get n + 4, m = 3, n 4 13, m = 5, n = 6 χ lc (P m P n ) = n, m = 5, n 8 3n + 5, m = 7, n 8 m + 9, m 9, n 10. Case 4. Let m and n be even. In this case, V (G 1 ) = V (G ) = mn. The graph P m P n has four vertices of each of degree one. Two of them are in G 1 and the other two vertices are in G. Assume that we start coloring the graph from the vertex w 11 in G 1, and assign color 1 to every vertex w ij when i and j are not both even. Thus, β(g 1 ) = mn 4 vertices are colored with color 1 and mn 4 vertices in G 1 remain uncolored. For color, assume that we start coloring the graph G from the vertex w 1, and consider the vertices w ij, where i {, 4,..., m }, j {1, 3,..., n 1} and w nl, where l {3, 5,..., n 1}. Thus, ( m ) n 1 + n = mn 4 1 vertices in G

8 568 C. ÇIFTÇI AND P. DÜNDAR are colored with color, and the number of the remaining uncolored vertices in G is mn Case 4.1. Let m = and n. Since P P n = Pn and χ lc (P n ) = 1 + n by [7], we have χ lc (P P n ) = n + χ lc (P n ) = n n = 3n + 1. Case 4.. Let m = 4 and n 4. Since m = 4, there is not any pair of vertices which is colored with color 3 and color 4. Hence, we have χ lc (P 4 P n ) = + mn + 1 = n + 3. Case 4.3. Let m 6 and n 6. For every vertex of degree 4 in G 1 which is not adjacent with each other κ(w ij, w kl ) 3 is satisfied, where i, k {, 4,..., m }, j, l {, 4,..., n }. Thus, ( m n m+n )( ) vertices are colored with color 3, and 1 vertices in G 1 remain uncolored. Since the number of internally disjoint paths between these remaining vertices is at most, all of them receive different colors. Hence, we have χ lc (G 1 ) = 1 + m+n. Case Let m = n = 6. Since there is not any pair of vertices in G which is colored with color 4, we have χ lc (P 6 P 6 ) = χ lc (G 1 ) mn = 18. Case Let m = 6, n 8. The number of internally disjoint paths between only two vertices in G is 4. Thus, the remaining mn 4 1 vertices receive different colors, and we have χ lc (P 6 P n ) = χ lc (G 1 ) + + mn 4 1 = n + 5. Case Let m 8 and n 8. Consider the vertices w ij, where i {5, 7,..., m 3}, j {, 4,..., n } and w kl, where k {3, m 1}, l {4, 6,..., n 4} in G. The number of internally disjoint paths between these vertices is 4. Thus, m 5 n n 5 ( ) + = mn 4 m+n 3 vertices receive color 4. The remaining m+n + 4 vertices receive different colors. Then χ lc (G ) = 6+ m+n and we have χ lc (P m P n ) = χ lc (G 1 )+χ lc (G ) = m+n+7. As a result, if m and n is even, we have 3n+1, m =, n n + 3, m = 5, n 4 χ lc (P m P n ) = 18, m = n = 6 n + 5, m = 6, n 6 m + n + 7, m 8, n 8. By summing up four cases we have the statement of Theorem. Theorem.. Let C m and P n be cycle and path of order m and n, respectively. Then,

9 LOCAL CONNECTIVE CHROMATIC NUMBER OF DIRECT PRODUCT OF , if m is odd m +, if m is even, n = χ lc (C m P n ) = m + 3, if m is even, n = 3 m + 4, if m is even, n 4 even or m is even, n 5 odd Proof. Since deg(w ij ) = deg(u i ) deg(v j ), we have κ(w ij, w kl ) min{w ij, w kl } 4, where w ij, w kl V (C m P n ), i, k {1,,..., m}, j, l {1,,..., n}. Thus, k i 1 for i 5, and the pair of vertices can be colored with the same color at most color 4. We have following two cases for coloring the graph C m P n. Figure. Local connective coloring of C 5 P 7 Case 1. Let m and n be odd or m be odd and n be even. By Theorem 1.1, since m is odd, the graph C m P n is connected, and by Theorem 1., C m P n is bipartite graph. Let C m P n = (V 1 V, E). Since κ(w ij, w kl ) 4, where w ij, w kl V (C m P n ), i, k {1,,..., m}, j, l {1,,..., n}, the vertices of V 1 and V can be colored with color 1 and color, respectively. Then we have χ lc (C m P n ) =. Case. Let m and n be even or m be even and n be odd. In this case, since cycles and paths are bipartite graphs, let C m = (V 0 V 1, E), P n = (W 0 W 1, F ). By Theorem 1.3 and Theorem 1., the graph C m P n is non-connected bipartite graph. Further, by Lemma 1. and 1.3, G 1 = ((V 0 W 0 ) (V 1 W 1 ), E F ) and G = ((V 0 W 1 ) (V 1 W 0 ), E F ) are two bipartite components of C m P n. Since G 1 is bipartite graph, χ lc (G 1 ) = by Case 1. Let s start coloring the graph G with color χ lc (G 1 ) + 1 = 3.

10 570 C. ÇIFTÇI AND P. DÜNDAR Since V (C m P n ) = V (C m ) V (P n ), it is obvious that V (C n ) = V (P m ) = mn. Case.1. Let n = and m be even. Since P = K, by Lemma 1.1 the graph C m K has exactly two connected components G 1 and G isomorphic to C m. Since G is regular graph, the number of internally disjoint paths between all two vertices in G is at most. Thus, all vertices of G receive different colors. Then, χ lc (C m P ) = χ lc (G 1 ) + χ lc (G ) = m +. Case.. Let n = 3 and m be even. In this case, the graph G has m(n ) = m vertices of degree 4, and these vertices are either in the vertex set V 0 W 1 or in V 1 W 0. That is, these vertices are not adjacent. Thus, m vertices are colored with color 3. The number of the remaining uncolored vertices is m. Since the degree of these remaining vertices is, these m vertices receive different colors. Thus, we have χ lc (C m P 3 ) = χ lc (G 1 ) + χ lc (G ) = m = m + 3. Case.3. Let m be even and n 4 even or m be even and n 5 odd. The graph C m P n has m(n ) vertices each of degree 4, and G has half of these vertices. The vertex sets W 0 and W 1 have n n and internal vertices of P n, respectively. Since κ(w ij, w kl ) 4, where w ij, w kl V (G ), we color m n vertices of G with color 3 and m n vertices of G with color 4. The remaining m vertices receive different colors. Thus, we have χ lc (C m P n ) = χ lc (G 1 ) + χ lc (G ) = + + m = m + 4. Theorem.3. Let C m and C n be two cycles of order m and n, respectively. Then,, if m is odd, n is even or m is even, n is odd χ lc (C m C n ) = 3, if m and n are odd 4, if m and n are even. Proof. By proof of Theorem 1.4, the graph C m C n is 4 regular graph. Then the number of internally disjoint paths between any two vertices in C m C n is at most 4. Hence, k i 1 for i 5. We have following three cases for coloring the graph C m C n. Case 1. Let m be odd and n be even or m be even and n be odd. It is known that if the order of a cycle is even, it is bipartite graph. Thus, by Theorem 1.1 and Theorem 1., the graph C m C n is bipartite connected graph. Since the number of internally disjoint paths between any two vertices in C m C n is at most 4, we have χ lc (C m C n ) =. Case. Let m and n be even. Since m and n are even, C m and C n are bipartite graphs. By Theorem 1., Theorem 1.3 and Lemma 1.3, the graph C m C n is bipartite non-connected graph

11 LOCAL CONNECTIVE CHROMATIC NUMBER OF DIRECT PRODUCT OF Figure 3. Local connective coloring of C 5 C 7 and has exactly two isomorpic connected components G 1 and G. Thus, these components are also bipartite graphs. Since G 1 is bipartite graph and the number of internally disjoint paths between any two vertices of G 1 is at most 4, we have χ lc (G 1 ) =. Let s start coloring the graph G with color χ lc (G 1 )+1 = 3. Since G is also bipartite graph and the number of internally disjoint paths between any two vertices of G is at most 4, the vertices of G receive color 3 and color 4. Hence, χ lc (C m C n ) = 4. Case 3. Let m and n be odd. Since m and n are odd, the graph C m C n is connected by Theorem 1.1. Assume that we start coloring the graph from the vertex w 11. By definition of direct product, the vertex w 1j can be colored with color 1 for j {1,,..., n}. Thus we color all non adjacent vertices w 1j, w 3j, w 5j,...,w (m )j with color 1. Since the vertices w j, w 4j, w 6j,...,w (m 1)j are not adjacent with each other, and the number of internally disjoint paths between them is at most 4, these vertices receive color. Then total n(m 1) vertices in C m C n are colored with color 1 and color. Hence, the vertex w mj remains uncolored and the number of its vertices is mn n(m 1) = n. Since the vertex w mj is adjacent to the vertices w 1j and w (m 1)j, the vertex w mj is colored with different color other than color 1 and color. Thus, all vertices in the graph C m C n are colored with three local connective colors. 3. Conclusion In this paper, we define a new type of graph coloring called local connective coloring. It is known that a communication network is fault-tolerant if it has alternative paths (internally disjoint paths) between vertices and the internally disjoint paths are used to transmit messages among vertices. Thus, we use the

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