Rainbow connection and forbidden subgraphs

A connected edge-colored graph G is rainbow-connected if any two distinct vertices of G are connected by a path whose edges have pairwise distinct colors; the rainbow connection number rc(G) of G is the minimum number of colors such that G is rainbow-connected. We consider families F of connected graphs for which there is a constant k(F) such that, for every connected F-free graph G, rc(G) <= diam(G) + k(F), where diam(G) is the diameter of G. In this paper, we give a complete answer for vertical bar F vertical bar is an element of {1, 2}. (C) 2014 Elsevier B.V. All rights reserved.