The $$(k,\ell )$$ ( k , ℓ ) -Rainbow Index of Random Graphs

Abstract

A tree in an edge-colored graph G is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers k, \(\ell \) with \(k\ge 3\), the \((k,\ell )\)-rainbow index\(rx_{k,\ell }(G)\) of G is the minimum number of colors needed in an edge-coloring of G such that for any set S of k vertices of G, there exist \(\ell \) internally disjoint rainbow trees connecting S. This concept was introduced by Chartrand et. al., and there have been very few known results about it. In this paper, we establish a sharp threshold function for \(rx_{k,\ell }(G_{n,p})\le k\) and \(rx_{k,\ell }(G_{n,M})\le k,\) respectively, where \(G_{n,p}\) and \(G_{n,M}\) are the usually defined random graphs.