Near-Optimal, Distributed Edge Colouring via the Nibble Method

We give a distributed randomized algorithm to edge colour a network. Let G be a graph with n nodes and maximum degree ∆. Here we prove: • If ∆ = Ω(log n) for some δ > 0 and λ > 0 is fixed, the algorithm almost always colours G with (1 + λ)∆ colours in time O(logn). • If s > 0 is fixed, there exists a positive constant k such that if ∆ = Ω(log n), the algorithm almost always colours G with ∆ +∆/ log n = (1 + o(1))∆ colours in time O(logn+ log n log logn). By “almost always” we mean that the algorithm may fail, but the failure probability can be made arbitrarily close to 0. The algorithm is based on the nibble method, a probabilistic strategy introduced by Vojtěch Rödl. The analysis makes use of a powerful large deviation inequality for functions of independent random variables.