The $(k,\ell )$-rainbow index for complete bipartite and multipartite 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 \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 results about it. In this paper, we investigate the $(k,\ell )$-rainbow index for complete bipartite graphs and complete multipartite graphs. Some asymptotic values of their $(k,\ell )$-rainbow index are obtained.

arXiv comment13 pages. arXiv admin note: substantial text overlap with arXiv:1212.6845

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