On the rainbow k-connectivity of complete 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 k-connectivity rck(G) of G is defined as the minimum integer j for which there exists a j-edge-coloring of G such that every two distinct vertices of G are connected by k internally disjoint rainbow paths. This paper is to investigate the rainbow k-connectivity of complete graphs. We improve the upper bound of f(k) from (k + 1) by Chartrand et al. to ck 3 2 + C(k), where c is a constant, C(k) = o(k 3 2 ), and f(k) is the integer such that if n ≥ f(k) then rck(Kn) = 2. Recently, we saw that Dellamonica et al. got the best possible upper bound 2k, which is linear in k. However, our proof is more structural or constructive.