Entropy compression method applied to graph colorings
Abstract
Based on the algorithmic proof of Lovász local lemma due to Moser and Tardos, the works of Grytczuk et al. on words, and Dujmović 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 socalled \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 $10$.
 Publication:

arXiv eprints
 Pub Date:
 June 2014
 arXiv:
 arXiv:1406.4380
 Bibcode:
 2014arXiv1406.4380G
 Keywords:

 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics
 EPrint:
 33 pages