Hardness and algorithms for rainbow connection

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 [6]. 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 2 > 0, a connected graph with minimum degree at least 2n has bounded rainbow connection, where the bound depends only on 2, 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.