On Rainbow-k-Connectivity of Random Graphs

A path in an edge-colored graph is called a rainbow path if the edges on it have distinct colors. For k ≥ 1, the rainbow-k-connectivity of a graph G, denoted rck(G), is the minimum number of colors required to color the edges of G in such a way that every two distinct vertices are connected by at least k internally vertex-disjoint rainbow paths. In this paper, we study rainbow-k-connectivity in the setting of random graphs. We show that for every fixed integer d ≥ 2 and every k ≤ O(log n), p = (log n)/n is a sharp threshold function for the property rck(G(n, p)) ≤ d. This substantially generalizes a result in [Y. Caro, A. Lev, Y. Roditty, Z. Tuza, and R. Yuster, On rainbow connection, Electron. J. Comb., 15, 2008], stating that p = √ logn/n is a sharp threshold function for the property rc1(G(n, p)) ≤ 2. As a byproduct, we obtain a polynomial-time algorithm that makes G(n, p) rainbow-k-connected using at most one more than the optimal number of colors with probability 1−o(1), for all k ≤ O(log n) and p = n for any constant ǫ ∈ [0, 1).