Sampling Proper Colorings on Line Graphs Using $(1+o(1))\Delta$ Colors

TL;DR

This paper proves that the Glauber dynamics efficiently samples proper colorings on line graphs with nearly optimal number of colors, using advanced matrix techniques to establish rapid mixing times.

Contribution

It introduces a new proof technique employing the matrix trickle-down theorem to show rapid mixing for coloring line graphs with slightly more than the maximum degree in colors.

Findings

Glauber dynamics mixes in O(n log n) time for q > (1+o(1))Δ

The proof leverages the matrix trickle-down theorem

Results improve understanding of sampling proper colorings on line graphs

Abstract

We prove that the single-site Glauber dynamics for sampling proper $q$-colorings mixes in $O_\Delta(n\log n)$ time on line graphs with $n$ vertices and maximum degree $\Delta$ when $q>(1+o(1))\Delta$. The main tool in our proof is the matrix trickle-down theorem developed by Abdolazimi, Liu and Oveis Gharan (FOCS, 2021).