Bounds for the rainbow connection number of graphs

Abstract

An edge-coloured graph G is rainbow-connected if any two vertices are connected by a path whose edges have distinct colours. The rain- bow 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. In this paper we show some new bounds for the rainbow connection number of graphs depending on the minimum degree and other graph parameters. Moreover, we discuss sharpness of some of these bounds.