An Optimal Decentralized (? + 1)-Coloring Algorithm

Consider the following simple coloring algorithm for a graph on n vertices. Each vertex chooses a color from {1, ..., ?(G) + 1} uniformly at random. While there exists a conflicted vertex choose one such vertex uniformly at random and recolor it with a randomly chosen color. This algorithm was introduced by Bhartia et al. [MOBIHOC\u2716] for channel selection in WIFI-networks. We show that this algorithm always converges to a proper coloring in expected O(n log ?) steps, which is optimal and proves a conjecture of Chakrabarty and de Supinski [SOSA\u2720]