Decay of correlation for edge colorings when $q>3\Delta$

TL;DR

This paper investigates the decay of correlation in proper q-edge colorings of graphs with maximum degree Δ, establishing conditions for coupling independence, strong spatial mixing, and weak spatial mixing.

Contribution

It proves coupling independence for q≥3Δ, strong spatial mixing on trees for q> (3+o(1))Δ, and characterizes weak spatial mixing thresholds as q≥2Δ-1.

Findings

Coupling independence holds when q≥3Δ.

Strong spatial mixing on trees for q> (3+o(1))Δ.

Weak spatial mixing on trees for q≥2Δ-1.

Abstract

We examine various perspectives on the decay of correlation for the uniform distribution over proper $q$-edge colorings of graphs with maximum degree $\Delta$. First, we establish the coupling independence property when $q\ge 3\Delta$ for general graphs. Together with the work of Chen et al. (2024), this result implies a fully polynomial-time approximation scheme (FPTAS) for counting the number of proper $q$-edge colorings. Next, we prove the strong spatial mixing property on trees, provided that $q> (3+o(1))\Delta$. The strong spatial mixing property is derived from the spectral independence property of a version of the weighted edge coloring distribution, which is established using the matrix trickle-down method developed in Abdolazimi, Liu and Oveis Gharan (FOCS, 2021) and Wang, Zhang and Zhang (STOC, 2024). Finally, we show that the weak spatial mixing property holds on trees with maximum degree $\Delta$ if and only if $q\ge 2\Delta-1$.