The $(k,\ell )$-rainbow index of random graphs

Abstract

A tree in an edge colored graph 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 \emph{$(k,\ell )$-rainbow index} $r{x}_{k,\ell }(G)$ 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 related results about it. In this paper, We establish a sharp threshold function for $r{x}_{k,\ell }(G_{n,p})\le k$ and $r{x}_{k,\ell }(G_{n,M})\le k,$ respectively, where $G_{n,p}$ and $G_{n,M}$ are the usually defined random graphs.

7 pages. arXiv admin note: substantial text overlap with arXiv:1212.6845, arXiv:1310.2783