An edge‐colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In the first result of this paper we prove that computing rc(G) is NP‐Hard solving an open problem from Caro et al. (Electron. J. Comb. 15, 2008, Paper R57). In fact, we prove that it is already NP‐Complete to decide if rc(G)=2, and also that it is NP‐Complete to decide whether a given edge‐colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every ϵ>0, a connected graph with minimum degree at least ϵ
n has bounded rainbow connection, where the bound depends only on ϵ, and a corresponding coloring can be constructed in polynomial time. Additional non‐trivial upper bounds, as well as open problems and conjectures are also presented.
