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

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