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

Abstract

Abstract We analyze Markov chains for generating a random k ‐coloring of a random graph G n , d / n . When the average degree d is constant, a random graph has maximum degree Θ (log n /log log n ), with high probability. We show that, with high probability, an efficient procedure can generate an almost uniformly random k ‐coloring when k = Θ (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. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006