Rainbow connection and minimum degree

An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. Krivelevich and Yuster have shown that a connected graph G with n vertices has rc(G) < 20n δ(G) [M. Krivelevich and R. Yuster, The rainbow connection of a graph is (at most) reciprocal to its minimum degree, Journal of Graph Theory 63 (2010) 185-191]. In this talk we prove that a connected graph G with n vertices has rc(G) < 4n δ(G)+1 + 4.