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Sampling Proper Colorings on Line Graphs Using $(1+o(1))Δ$ Colors
Yulin Wang, Chihao Zhang, Zihan Zhang · 2023-07-16 · via math.PR updates on arXiv.org

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

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