On 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 number 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 this paper we prove several non-trivial upper bounds for rc(G), as well as determine sufficient conditions that guarantee rc(G) = 2. Among our results we prove that if G is a connected graph with n vertices and with minimum degree 3 then rc(G) < 5n/6, and if the minimum degree is δ then rc(G) ≤ ln δ δ n(1 + oδ(1)). We also determine the threshold function for a random graph to have rc(G) = 2 and make several conjectures concerning the computational complexity of rainbow connection. ∗Research supported in part by the Hungarian Scientific Research Fund, OTKA grant T-049613 the electronic journal of combinatorics 15 (2008), #R57 1