  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).
