Spatial Mixing of Coloring Random Graphs

TL;DR

This paper proves that proper q-colorings of random graphs G(n, d/n) exhibit strong spatial mixing under certain conditions, a novel result for graphs with unbounded maximum degree, using a block-wise correlation decay approach.

Contribution

It establishes the first strong spatial mixing result for colorings of graphs with unbounded maximum degree, specifically for G(n, d/n), using a new block-wise correlation decay method.

Findings

Strong spatial mixing holds for q ≥ αd + β with α > 2.

Colorings exhibit correlation decay with high probability.

Introduces a block-wise decay analysis for unbounded degree graphs.

Abstract

We study the strong spatial mixing (decay of correlation) property of proper $q$-colorings of random graph $G(n, d/n)$ with a fixed $d$. The strong spatial mixing of coloring and related models have been extensively studied on graphs with bounded maximum degree. However, for typical classes of graphs with bounded average degree, such as $G(n, d/n)$, an easy counterexample shows that colorings do not exhibit strong spatial mixing with high probability. Nevertheless, we show that for $q\ge\alpha d+\beta$ with $\alpha>2$ and sufficiently large $\beta=O(1)$, with high probability proper $q$-colorings of random graph $G(n, d/n)$ exhibit strong spatial mixing with respect to an arbitrarily fixed vertex. This is the first strong spatial mixing result for colorings of graphs with unbounded maximum degree. Our analysis of strong spatial mixing establishes a block-wise correlation decay instead of the standard point-wise decay, which may be of interest by itself, especially for graphs with unbounded degree.