On strong rainbow connection number

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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 src(G), is the minimum number of colors that are needed in order to make G strong rainbow connected. In this paper, we first investigate the graphs with large strong rainbow connection numbers. Chartrand et al. obtained that G is a tree if and only if src(G)=m, we will show that src(G)≠ m-1, so G is not a tree if and only if src(G)≤ m-2, where m is the number of edge of G. Furthermore, we characterize the graphs G with src(G)=m-2. We next give a sharp upper bound for src(G) according to the number of edge-disjoint triangles in graph G, and give a necessary and sufficient condition for the equality.