Rainbow Connection Of The Join Of Two Paths

Abstract

An edge-colored graph \(G\) is (strong) rainbow connected if any two vertices are connected by a (geodesic) path whose edges have distinct colors. The (strong) rainbow connection number of a connected graph \(G\), denoted by \(\mathrm{src}(G)\) (resp. \(\mathrm{rc}(G)\)), is the smallest number of colors that are needed in order to make \(G\) (strong) rainbow connected. The join \(P_m \vee P_n\) of \(P_m\) and \(P_n\) is the graph consisting of \(P_m\cup P_n\), and all edges between every vertex of \(P_m\) and every vertex of \(P_n\), where \(P_m\) (resp. \(P_n\)) is a path of \(m\) (resp. \(n\)) vertices. In this paper, the precise values of \(\mathrm{rc}(P_m \vee P_n)\) and \(\mathrm{src}(P_m \vee P_n)\) are given for any positive integers \(m\) and \(n\).