On forbidden subgraphs and rainbow connection in graphs with minimum degree 2

A connected edge-colored graph GG is said to be rainbow-connected if any two distinct vertices of GG are connected by a path whose edges have pairwise distinct colors, and the rainbow connection number rc(G) of GG is the minimum number of colors that can make GG rainbow-connected. We consider families FF of connected graphs for which there is a constant kFkF such that every connected FF-free graph GG with minimum degree at least 2 satisfies rc(G)≤diam(G)+kF, where diam(G) is the diameter of GG. In this paper, we give a complete answer for |F|=1|F|=1, and a partial answer for |F|=2|F|=2.