Randomly coloring random graphs

Abstract

Abstract We consider the problem of generating a coloring of the random graph 𝔾 n , p uniformly at random using a natural Markov chain algorithm: the Glauber dynamics. We assume that there are β Δ colors available, where Δ is the maximum degree of the graph, and we wish to determine the least β = β ( p ) such that the distribution is close to uniform in O ( n log n ) steps of the chain. This problem has been previously studied for 𝔾 n , p in cases where n p is relatively small. Here we consider the “dense” cases, where n p ε [ ω ln n , n ] and ω = ω ( n ) → ∞. Our methods are closely tailored to the random graph setting, but we obtain considerably better bounds on β ( p ) than can be achieved using more general techniques. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009