On the rainbow connection for some corona graphs

Abstract

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is rainbow connected if there exists a rainbow path between every pair of vertices. The rainbow connectivity of a graph G, denoted by rc(G) is the smallest number of colors required to edge color the graph such that the graph is rainbow connected.In this note, we determine the exact values of rc (G • H) where G or H are complete graph K n , path P n , tree T n , or wheel W n with n is an integer. Mathematics Subject Classification: 05C15, 05C40

The rainbow connection of windmill and corona graph

“…Then, K i m is the symbol of the i'th complete graph. Case 1 n = 3, to show that rc(K (3) m ) ≤ 3, we provide a rainbow 3-coloring c :…”

“…This implies that all the edges of the i'th complete graph are colored with the color i − 1. Therefore, rc(K (3) m ) = 3. Case 2 n > 3, we define a 3-coloring c : E(K (n) m ) → {0, 1, 2}, let any three of the complete graphs be colored 0, 1, 2 respectively.…”

“…An edge-coloring graph G is rainbow connected if any two vertices are connected by a rainbow path. The minimum k for which there exists a rainbow k-coloring of the edges of G is the rainbow connection number rc(G) of G. For two vertices u and v of G, a rainbow u − v geodesic in G is a rainbow u−v path of length d(u, v), where d (u, v) is the distance between u and v. The graph G is strongly rainbow-connected if G contains a rainbow u − v geodesic for every two vertices u and v of G. The minimum k for which there exists a coloring c of the edges of G such that G is strongly rainbow-connected is the strong rainbow connection number src(G) of G. Thus, rc(G)≤src(G) for every connected graph G. Furthermore, if G is a nontrivial connected graph of size m whose diameter denoted by diam(G), then diam(G) ≤ rc(G) ≤ src(G) ≤ m. The rainbow connection of fan, sun and some corona graphs have been studied in [2,3]. There have been some results on the (strong) rainbow connection numbers of graphs (see [1,4]).…”

Rainbow connection numbers of some graphs

“…Chartrand et al obtained that rc(G) = 1 if and only if G is complete, and that rc(G) = m if and only if G is a tree, as well as that a cycle with k > 3 vertices has rainbow connection number k 2 , a triangle has rainbow connection number 1 ( [1]). Also notice that, clearly, rc(G) ≥ diam(G) where diam(G) denotes the diameter of G. Furthermore, Dewi Estetikasari and Syafrizal Sy [2] determined the exact values of rainbow connection for some corona graph.…”

Rainbow connection number of the thorn graph

“…The rainbow connection of fan, sun and some corona graphs have been studied in [2,3]. There have been some results on the (strong) rainbow connection numbers of graphs (see [1,4]).…”