Entropy compression method applied to graph colorings

Based on the algorithmic proof of Lovász local lemma due to Moser and Tardos, 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 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 ∆ has an acyclic chromatic number at most 3 2 ∆ 4 3 +O(∆), and a non-repetitive chromatic number at most ∆ + 1.89∆ 5 3 + O(∆ 4 3 ). Also every planar graph with maximum degree ∆ has a facial Thue chromatic number at most ∆+ O(∆ 1 2 ) and a facial Thue chromatic index at most 10.