On the Facial Thue Choice Index via Entropy Compression

Abstract

A sequence is nonrepetitive if it contains no identical consecutive subsequences. An edge coloring of a path is nonrepetitive if the sequence of colors of its consecutive edges is nonrepetitive. By the celebrated construction of Thue, it is possible to generate nonrepetitive edge colorings for arbitrarily long paths using only three colors. A recent generalization of this concept implies that we may obtain such colorings even if we are forced to choose edge colors 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', Soták, and Škrabul'áková proved that for each plane graph, eight colors are sufficient to provide an edge coloring so that every facial path is nonrepetitively colored. In this article, 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 Lovász Local Lemma). Our approach is based on the Moser–Tardos entropy‐compression method and its recent extensions by Grytczuk, Kozik, and Micek, and by Dujmović, Joret, Kozik, and Wood.

Focused Stochastic Local Search and the Lovász Local Lemma

“…In these algorithms the state space Ω is the set of all partial assignments that do not violate any constraint, while there is one flaw for each unassigned variable: if fixing a flaw (i.e., assigning a variable) causes one or more constraints to become violated, the algorithm backtracks by unassigning not only the last variable set but several more-typically all variables involved in some violated constraint. Examples of such algorithms include [36,30,27,34,35,59,61,21,53,13,19]. Our sharpened theorem immediately provides a unified and greatly simplified analysis of such algorithms (see Section 6 for examples).…”

Planar graphs have bounded nonrepetitive chromatic number

“…The best known upper bound is O(log n) where n is the number of vertices, due to Dujmović, Frati, Joret, and Wood [16]. Note that several works have studied colourings of planar graphs in which only facial paths are required to be nonrepetitively coloured [4,8,33,34,44,45,48].…”

A note on acyclic vertex-colorings

“…All these ideas inspired new adaptations and more efficient uses of the essence of the Local Lemma to tackle various combinatorial questions, in particular graph colouring problems [4,6,9,14,15] and problems related to pattern avoidance [13]. We draw inspiration from the original work of Alon, McDiarmid & Reed [2] and a recent result of Esperet & Parreau [6] to establish the following upper bound.…”