Progress on Rainbow Connection

An edge-coloured graph G is called rainbow-connected if any two vertices are connected by a path whose edges have different colours. This concept of rainbow connection in graphs was recently introduced by Chartrand et al. in [4]. 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. An easy observation is that if G has n vertices then rc(G) ≤ n−1, since one may colour the edges of a given spanning tree of G with different colours, and colour the remaining edges with one of the already used colours. Chartrand et al. computed the precise rainbow connection number of several graph classes including complete multipartite graphs [4]. The rainbow connection number has been studied for further graph classes in [3] and for graphs with fixed minimum degree in ([3], [7], [9]).