The rainbow $k$-connectivity of two classes of graphs

TL;DR

This paper investigates the rainbow k-connectivity of complete multipartite graphs, establishing bounds on the minimum number of colors needed for certain connectivity properties, and improving existing bounds for complete graphs.

Contribution

It determines conditions under which the rainbow k-connectivity of certain graphs equals 2, and improves bounds on the function f(k) for complete graphs.

Findings

For large enough number of parts, the rainbow k-connectivity is 2.

Improved upper bounds on the function f(k) from quadratic to sub-quadratic in k.

Established thresholds for the number of parts in multipartite graphs to achieve rainbow 2-connectivity.

Abstract

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of $G$ are colored the same. For a $\kappa$-connected graph $G$ and an integer $k$ with $1\leq k\leq \kappa$, the rainbow $k$-connectivity $rc_k(G)$ of $G$ is defined as the minimum integer $j$ for which there exists a $j$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by $k$ internally disjoint rainbow paths. Let $G$ be a complete $(\ell+1)$-partite graph with $\ell$ parts of size $r$ and one part of size $p$ where $0\leq p <r$ (in the case $p=0$, $G$ is a complete $\ell$-partite graph with each part of size $r$). This paper is to investigate the rainbow $k$-connectivity of $G$. We show that for every pair of integers $k\geq 2$ and $r\geq 1$, there is an integer $f(k,r)$ such that if $\ell\geq f(k,r)$, then $rc_k(G)=2$. As a consequence, we improve the upper bound of $f(k)$ from $(k+1)^2$ to $ck^{{3/2}}+C$, where $0<c<1$, $C=o(k^{{3/2}})$, and $f(k)$ is the integer such that if $n \geq f(k)$ then $rc_k(K_n)=2$.