Rainbow connection and forbidden subgraphs

A connected edge-colored graph GG is rainbow-connected if any two distinct vertices of GG are connected by a path whose edges have pairwise distinct colors; the rainbow connection number rc(G) of GG is the minimum number of colors such that GG is rainbow-connected. We consider families FF of connected graphs for which there is a constant kFkF such that, for every connected FF-free graph GG, rc(G)≤diam(G)+kF, where diam(G) is the diameter of GG. In this paper, we give a complete answer for |F|∈{1,2}|F|∈{1,2}.