Randomly coloring sparse random graphs with fewer colors than the maximum degree

We analyze Markov chains for generating a random k-coloring of a random graph Gn,d/n. When the average degree d is constant, a random graph has maximum degree log n/ log log n, with high probability. We efficiently generate a random k-coloring when k = Ω(log log n/ log log log n), i.e., with many fewer colors than the maximum degree. Previous results hold for a more general class of graphs, but always require more colors than the maximum degree.For k ≥ (lnn) we can also show that Glauber Dynamics mixes in polynomial time.