On the strong rainbow connection of a graph∗

A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. For any 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 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 that are needed in order to make G strongly rainbow connected. In this paper, we first give a sharp upper bound for src(G) in terms of the number of edge-disjoint triangles in a graph G, and give a necessary and sufficient condition for the equality. We next investigate the graphs with large strong rainbow connection numbers. Chartrand et al. obtained that src(G) = m if and only if G is a tree, we will show that src(G) 6= m− 1, and characterize the graphs G with src(G) = m − 2 where m is the number of edges of G.