On the Rainbow Connection Number of Graphs

Abstract

Let \(G = (V, E)\), with \(|V| = n\), be a simple connected graph. An edge-colored graph \(G\) is rainbow edge-connected if any two vertices are connected by a path whose edges are colored by distinct colors. The rainbow connection number of a connected graph \(G\), denoted by \(rc(G)\), is the smallest number of colors that are needed in order to make \(G\) rainbow edge-connected. In this paper, we obtain tight bounds for \(rc(G)\). We use our results to generalize previous results for graphs with \(\delta(G) \geq 3\).