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 kF such that, for every connected F-free graph G, rc(G) ≤ diam(G)+ kF , where diam(G) is the diameter of G. In this paper, we give a complete answer for |F| ∈ {1, 2}.