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 ≤ k ≤ κ, the rainbow kconnectivity rck(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 Kr,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 Kr,r is 3 for every integer r ≥ g(k). This short note is to solve this question by showing that rck(Kr,r) = 3 for every integer r ≥ 2k⌈ k 2⌉, where k ≥ 2 is a positive integer.