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

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 $κ$-connected graph $G$ and an integer $k$ with $1\leq k\leq κ$, 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 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 "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 $r\geq g(k)$. This short note is to solve this question by showing that $rc_k(K_{r,r})=3$ for every integer $r\geq 2k\lceil\frac{k}{2}\rceil$, where $k\geq 2$ is a positive integer.