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

All graphs considered in this paper are simple, finite and undirected. Let G be a nontrivial connected graph with an edge coloring c : E(G) → {1, 2, • • • , k}, k ∈ N, where adjacent edges may be colored the same. A path of G is called rainbow if no two edges of it are colored the same. A well-known result shows that in every κ-connected graph G with κ ≥ 1, there are k internally disjoint uv paths connecting any two distinct vertices u and v for every integer k with 1 ≤ k ≤ κ. Chartrand et al. [2] defined the rainbow k-connectivity rc k (G) of G, which is the minimum integer j for which there exists a j-edge-coloring of G such that for any two distinct vertices u and v of G, there exist at least k internally disjoint uv rainbow paths. The concept of rainbow k-connectivity has applications in transferring information of high security in communication networks. For details we refer to [2] and [3]. In [2], Chartrand et al. studied the rainbow k-connectivity of the complete graph K n for various pairs k, n of integers. It was shown in [2] that for every integer k ≥ 2, there exists an integer f (k) such that rc k (K n ) = 2 for every integer n ≥ f (k). In [4], We improved the upper bound of f (k) from (k + 1) 2 to ck ). Chartrand et al. in [2] also investigated the rainbow k-connectivity of r-regular complete bipartite graphs for some pairs k, r of integers with 2 ≤ k ≤ r, and they showed that for every integer k ≥ 2, there exists an integer r such that rc k (K r,r ) = 3. However, they could not show a similar result as for complete graphs, and therefore they left an open question: For every integer k ≥ 2, determine an integer (function) g(k), for which rc k (K r,r ) = 3 for every integer r ≥ g(k), that is, the rainbow k-connectivity of the complete bipartite graph K r,r is essentially 3. This short note is to solve this question by showing that rc k (K r,r ) = 3 for every integer r ≥ 2k⌈ k 2 ⌉. We use a method similar to but more complicated than the proof of Theorem 2.3 in [2]. For notation and terminology not defined here, we refer to [1]. In [2], the authors derived the following results: