The Rainbow Connectivities of Small Cubic Graphs

Abstract

A path \(P\) in an edge-colored graph (not necessarily a proper edge-coloring) is a rainbow path if no two edges of \(P\) are assigned the same color. For a connected graph \(G\) with connectivity \(\kappa(G)\) and an integer \(k\) with \(1 \leq k \leq \kappa(G)\), the rainbow \(k\)-connectivity \(rc_k(G)\) of \(G\) is the minimum number of colors needed in an edge-coloring of \(G\) such that every two distinct vertices \(u\) and \(v\) of \(G\) are connected by at least \(k\) internally disjoint \(u-v\)rainbow paths. In this paper, the rainbow \(2\)-connectivity of the Petersen graph as well as the rainbow connectivities of all cubic graphs of order \(8\) or less are determined.