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=logn+omega/n where omega = omega(n) -> infinity and omega = o(logn) 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) similar to max{Z(1), diam(G)} with high probability (whp). Here Z(1) is the number of vertices in G whose degree equals 1 and the diameter of G is asymptotically equal to log n/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(log(2 theta r) n) where theta(r) = log(r-1)/log(r-2) when r >= 4 and rc(G) = O(log(4)n) whp when r = 3.