Rainbow Connectivity of Sparse Random Graphs

Abstract

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 $p=\frac{\log n+\omega }{n}$ where ω = ω(n) → ∞ and ω = 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 $rc(G) \sim \max\left\{Z_1,diameter(G)\right\}$ 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 $\frac{ \log{n}}{\log{\log{n}}}$ whp. Finally, we prove that the rainbow connectivity rc(G) of the random r-regular graph G = G(n,r) satisfies rc(G) = O(log² n) whp.