Entropy compression method applied to graph colorings

TL;DR

This paper introduces a generalized entropy compression framework to improve upper bounds on various graph chromatic numbers, leveraging algorithmic and combinatorial techniques.

Contribution

It develops a more general entropy compression framework and provides tighter bounds on chromatic numbers for graphs, including planar graphs.

Findings

Acyclic chromatic number of any graph with maximum degree Δ is at most (3/2)Δ^{4/3} + O(Δ).

Facial Thue choice number for planar graphs with maximum degree Δ is at most Δ + O(Δ^{1/2}).

Facial Thue choice index for planar graphs is at most 10.

Abstract

Based on the algorithmic proof of Lov\'asz local lemma due to Moser and Tardos, the works of Grytczuk et al. on words, and Dujmovi\'c et al. on colorings, Esperet and Parreau developed a framework to prove upper bounds for several chromatic numbers (in particular acyclic chromatic index, star chromatic number and Thue chromatic number) using the so-called \emph{entropy compression method}. Inspired by this work, we propose a more general framework and a better analysis. This leads to improved upper bounds on chromatic numbers and indices. In particular, every graph with maximum degree $\Delta$ has an acyclic chromatic number at most $\frac{3}{2}\Delta^{\frac43} + O(\Delta)$. Also every planar graph with maximum degree $\Delta$ has a facial Thue choice number at most $\Delta + O(\Delta^\frac 12)$ and facial Thue choice index at most $10$.