Near-optimal distributed edge coloring

Abstract

We give a distributed randomized algorithm to edge color a network. Given a graph G with n nodes and maximum degree A, the algorithm,- For any fixed λ > 0, colours G with (1+λ)Δ colours in time O(log n).- For any fixed positive integer s, colours G with Δ +Δ/(log Δ)s =(1+o(1))Δ colours in time O(log n + logs Δ log log Δ).Both results hold with probability arbitrarily close to 1 as long as Δ(G) =Ω(logl+d n), for some d > 0. The algorithm is based on the RSdl Nibble, a probabilistic strategy introduced by Vojtech Rödl. The analysis involves a certain quasi-random phenomenon involving sets at the vertices of the graph.