Randomly coloring regular bipartite graphs and graphs with bounded common neighbors

Abstract

Let G be an n-node graph with maximum degree Δ. The Glauber dynamics for G, defined by Jerrum, is a Markov chain over the k-colorings of G. Many classes of G on which the Glauber dynamics mixes rapidly have been identified. Recent research efforts focus on the important case that Δ ≥ d log 2 n holds for some sufficiently large constant d. We add the following new results along this direction, where ε can be any constant with 0 < ε < 1. - Let α ≈ 1.645 be the root of (1 - e -1/x)2 + 2xe-1/x = 2. If G is regular and bipartite and k ≥ (α+ε)Δ, then the mixing time of the Glauber dynamics for G is O(n log n). - Let β ≈ 1.763 be the root of x = e1/x. If the number of common neighbors for any two adjacent nodes of G is at most ε1.5Δ/360e and k ≥ (1 + ε)βΔ, then the mixing time of the Glauber dynamics is O(n log n). © Springer-Verlag Berlin Heidelberg 2012.