Strong spatial mixing of $q$-colorings on Bethe lattices

TL;DR

This paper proves strong spatial mixing for q-colorings on Bethe lattices and binary trees, establishing conditions on q relative to the lattice degree, which advances understanding of coloring problems in graph theory.

Contribution

It establishes the conditions under which strong spatial mixing occurs for q-colorings on Bethe lattices and binary trees, using analysis of the sum-product algorithm.

Findings

Strong spatial mixing holds for q ≥ 1 + ⌈1.764b⌉ on (b+1)-regular Bethe lattices.

Strong spatial mixing holds for q=4 on binary trees.

The analysis uses the sum-product algorithm to establish mixing properties.

Abstract

We investigate the problem of strong spatial mixing of $q$-colorings on Bethe lattices. By analyzing the sum-product algorithm we establish the strong spatial mixing of $q$-colorings on $(b+1)$-regular Bethe lattices, for $q \geq 1+\lceil 1.764b \rceil$. We also establish the strong spatial mixing of $q$-colorings on binary trees, for $q=4$.