On the facial Thue choice index via entropy compression

TL;DR

This paper improves bounds on list edge colourings of plane graphs to ensure nonrepetitive facial paths, using entropy compression methods to reduce the list size from 291 to 12.

Contribution

It demonstrates that list sizes of at least 12 are sufficient for nonrepetitive facial path colourings in plane graphs, improving previous bounds significantly.

Findings

Reduced list size bound from 291 to 12.

Applied entropy compression method to graph colouring.

Established new theoretical bounds for nonrepetitive facial path colourings.

Abstract

A sequence is nonrepetitive if it contains no identical consecutive subsequences. An edge colouring of a path is nonrepetitive if the sequence of colours of its consecutive edges is nonrepetitive. By the celebrated construction of Thue, it is possible to generate nonrepetitive edge colourings for arbitrarily long paths using only three colours. A recent generalization of this concept implies that we may obtain such colourings even if we are forced to choose edge colours from any sequence of lists of size 4 (while sufficiency of lists of size 3 remains an open problem). As an extension of these basic ideas, Havet, Jendrol', Sot\'ak and \v{S}krabul'\'akov\'a proved that for each plane graph, 8 colours are sufficient to provide an edge colouring so that every facial path is nonrepetitively coloured. In this paper we prove that the same is possible from lists, provided that these have size at least 12. We thus improve the previous bound of 291 (proved by means of the Lov\'asz Local Lemma). Our approach is based on the Moser-Tardos entropy-compression method and its recent extensions by Grytczuk, Kozik and Micek, and by Dujmovi\'c, Joret, Kozik and Wood.