Characterize graphs with rainbow connection numbers m-2 and m-3

A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph G is rainbow connected if there is a rainbow path connecting any two vertices, and the rainbow connection number of G, denoted by rc(G), is the minimum number of colors that are needed in order to make G rainbow connected. Chartrand et al. showed that G is a tree if and only if rc(G) = m, and it is easy to see that G is not a tree if and only if rc(G) ≤ m − 2, where m is the number of edges of G. So an interesting problem arises: Characterize the graphs G with rc(G) = m− 2. In this paper, we resolve this problem. Furthermore, we also characterize the graphs G with rc(G) = m− 3.