On the rainbow connection number of graphs

Let G = (V, E), vertical bar V vertical bar = n, be a simple connected graph. An edge-colored graph G is rainbow edge-connected if any two vertices are connected by a path whose edges are colored by distinct colors. The rainbow connection number of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow edge connected. In this paper we obtain tight bounds for rc(G). We use our results to generalize previous results for graphs with delta(G) >= 3.