Note on the Rainbow $k$-Connectivity of Regular Complete Bipartite Graphs

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 the path are colored the same. For a $k$-connected graph $G$ and an integer $k$ with $1\le k\le \kappa$, the rainbow $k$-connectivity $r{c}_k(G)$ of $G$ is defined as the minimum integer $j$ for which there exists a $j$-edge-coloring of $G$ such that any two distinct vertices of $G$ are connected by $k$ internally disjoint rainbow paths. Denote by $K_{r,r}$ an $r$-regular complete bipartite graph. Chartrand et al. in in “G. Chartrand, G.L. Johns, K.A.McKeon, P. Zhang, The rainbow connectivity of a graph, Networks $54(2009),75-81”$ left an open question of determining an integer $g(k)$ for which the rainbow $k$-connectivity of $K_{r,r}$ is $3$ for every integer $r\ge g(k)$. This short note is to solve this question by showing that $r{c}_k(K_{r,r})=3$ for every integer $r\ge 2k\lceil \frac{k}{2}\rceil$, where $k\ge 2$ is a positive integer.