The rainbow connection of fan and sun

Abstract

A path in an edge−colored graph is said to be a rainbow path if every edge in the path has different color. An edge colored graph is rainbow connected if there exists a rainbow path between every pair of vertices. The rainbow connection of a graph G, denoted by rc(G), is the smallest number of colors required to color the edges of graph such that the graph is rainbow connected. Given two arbitrary vertices u and v in 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 there exists a rainbow u−v geodesic for any two vertices u and v in G. The strong rainbow connection number of G, denoted by src(G), is the minimum number of colors needed to make G strongly rainbow connected. In this paper we determine the exact values of rc(G) and src(G) where G are Fn and Sm with n + 1 and 2m vertices, respectively.