Rainbow Connection of Sparse Random Graphs

An edge colored graph G is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this work we study the rainbow connectivity of binomial random graphs at the connectivity threshold p = log n+ω n where ω = ω(n) → ∞ and ω = o(log n) and of random r-regular graphs where r ≥ 3 is a fixed integer. Specifically, we prove that the rainbow connectivity rc(G) of G = G(n, p) satisfies rc(G) ∼ max {Z1, diameter(G)} with high probability (whp). Here Z1 is the number of vertices in G whose degree equals 1 and the diameter of G is asymptotically equal to logn log log n whp. Finally, we prove that the rainbow connectivity rc(G) of the random r-regular graph G = G(n, r) whp satisfies rc(G) = O(logr n) where θr = log(r−1) log(r−2) when r ≥ 4 and rc(G) = O(log n) whp when r = 3.