The rainbow k-connectivity of two classes of 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 G 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-edgecoloring of G such that every two distinct vertices of G are connected by k internally disjoint rainbow paths. Let G be a complete (l + 1)-partite graph with l parts of size r and one part of size p where 0 ≤ p < r (in the case p = 0, G is a complete l-partite graph with each part of size r). This paper is to investigate the rainbow k-connectivity of G. We show that for every pair of integers k ≥ 2 and r ≥ 1, there is an integer f(k, r) such that if l ≥ f(k, r), then rck(G) = 2. As a consequence, we improve the upper bound of f(k) from (k + 1)2 to ck 3 2 + C, where 0 < c < 1, C = o(k 3 2 ), and f(k) is the integer such that if n ≥ f(k) then rck(Kn) = 2.