Rainbow Connection for Complete Multipartite Graphs

TL;DR

This paper determines the minimum size of parts in complete multipartite graphs needed to guarantee low rainbow k-connection numbers, solving an open problem and providing new bounds for such graphs.

Contribution

It establishes an exact formula for the function f(k,t) related to rainbow k-connection in complete multipartite graphs, answering a question posed by Fujita, Liu, and Magnant.

Findings

f(k,t) = ceil(2k/(t-1)) for all k≥2, t≥2

Provided conditions for rainbow k-connection numbers to be at most 3 or 2

Solved an open problem in the theory of rainbow connectivity in multipartite graphs

Abstract

A path in an edge-colored graph is said to be rainbow if no color repeats on it. An edge-colored graph is said to be rainbow $k$-connected if every pair of vertices is connected by $k$ internally disjoint rainbow paths. The rainbow $k$-connection number $\mathrm{rc}_k(G)$ is the minimum number of colors $\ell$ such that there exists a coloring with $\ell$ colors that makes $G$ rainbow $k$-connected. Let $f(k,t)$ be the minimum integer such that every $t$-partite graph with part sizes at least $f(k,t)$ has $\mathrm{rc}_k(G) \le 4$ if $t=2$ and $\mathrm{rc}_k(G) \le 3$ if $t \ge 3$. Answering a question of Fujita, Liu and Magnant, we show that \[ f(k,t) = \left\lceil \frac{2k}{t-1} \right\rceil \] for all $k\geq 2$, $t\geq 2$. We also give some conditions for which $\mathrm{rc}_k(G) \le 3$ if $t=2$ and $\mathrm{rc}_k(G) \le 2$ if $t \ge 3$.