Fast Distributed Algorithms for Brooks–Vizing Colorings - Grable, David A; Panconesi, Alessandro (Journal of algorithms, 2000)
Let G be a Δ-regular graph with n vertices and girth at least 4 such that Δ⪢logn. We give very simple, randomized, distributed algorithms for vertex coloring G with Δ/k colors in O(k+logn) communication rounds, where k=O(logΔ). The algorithm may fail or exceed the above running time, but the probability that this happens is o(1), a quantity that goes to zero as n grows. The probabilistic analysis relies on a powerful generalization of Azuma's martingale inequality that we dub the Method of Bounded Variances.