Characterizing forbidden pairs for rainbow connection in graphs with minimum degree 2

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 that are needed in order to make G rainbow connected. In this paper, we complete the discussion of pairs (X,Y ) of connected graphs for which there is a constant kXY such that, for every connected (X,Y )-free graph G with minimum degree at least 2, rc(G) ≤ diam(G) + kXY (where diam(G) is the diameter of G), by giving a complete characterization. In particular, we show that for every connected (Z3, S3,3,3)free graph G with δ(G) ≥ 2, rc(G) ≤ diam(G) + 156, and, for every connected (S2,2,2, N2,2,2)-free graph G with δ(G) ≥ 2, rc(G) ≤ diam(G) + 72.