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 ≥ dlog₂ 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 + 2x e ^− 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(nlogn).

• Let β ≈ 1.763 be the root of x = e ^1/x. If the number of common neighbors for any two adjacent nodes of G is at most $\frac{\epsilon^{1.5}\Delta}{360e}$ and k ≥ (1 + ε)βΔ, then the mixing time of the Glauber dynamics is O(nlogn).