The $(k,\ell)$-rainbow index for complete bipartite and multipartite graphs

TL;DR

This paper studies the minimum number of colors needed to ensure multiple disjoint rainbow trees connect any set of vertices in complete bipartite and multipartite graphs, extending the understanding of rainbow connectivity.

Contribution

It introduces new bounds and asymptotic values for the $(k, ext{ell})$-rainbow index in complete bipartite and multipartite graphs, a topic with limited prior research.

Findings

Derived asymptotic values for the $(k, ext{ell})$-rainbow index.

Established bounds for complete bipartite graphs.

Extended results to complete 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 $k\geq 3$, the \emph{$(k,\ell)$-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 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.