Randomly colouring random graphs

We consider the problem of generating a coloring of the random graph _n,p uniformly at random using a natural Markov chain algorithm: the Glauber dynamics. We assume that there are βΔ 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 _n,p in cases where np is relatively small. Here we consider the “dense” cases, where np 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.