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

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