Highly nonrepetitive sequences: Winning strategies from the local lemma

Pegden, Wesley

doi:10.1002/rsa.20354

Cited by 28 publications

(31 citation statements)

References 22 publications

(44 reference statements)

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“…This was followed ba a series of improvements, see . Finally, using the so called Lefthanded Local Lemma due to Pegden , Grytczuk et al. proved that lists of size 4 are sufficient (while the question whether the lists of size 3 would suffice remains wide open).…”

Section: Introductionmentioning

confidence: 99%

On the Facial Thue Choice Index via Entropy Compression

Przybyło

2013

Journal of Graph Theory

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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.

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Section: Introductionmentioning

confidence: 99%

On the Facial Thue Choice Index via Entropy Compression

Przybyło

2013

Journal of Graph Theory

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“…There have been quite a number of results regarding the winning strategies that Ann should employ depending on the possible length of the repeating sequence [2,4,1], most of which proved only the existence of such a strategy. However, when considering the case of sequences of length 2 or longer (nontrivial repetitions), which is the basic case given the fact that repetitions of length 2 are already created by Ben whenever he copies the symbol that Ann has previously added, deterministic approaches of such an algorithm have been explicitly provided for an alphabet of size 37 in [4] and later on improved to an alphabet of size 9 in [1].…”

Section: Preliminariesmentioning

confidence: 99%

“…However, when considering the case of sequences of length 2 or longer (nontrivial repetitions), which is the basic case given the fact that repetitions of length 2 are already created by Ben whenever he copies the symbol that Ann has previously added, deterministic approaches of such an algorithm have been explicitly provided for an alphabet of size 37 in [4] and later on improved to an alphabet of size 9 in [1].…”

Section: Preliminariesmentioning

confidence: 99%

A note on Thue games

Mercaş

Nowotka

2017

Information Processing Letters

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In this work we improve on a result from [1]. In particular, we investigate the situation where a word is constructed jointly by two players who alternately append letters to the same end of the construction. One of the players (Ann) tries to avoid (non-trivial) repetitions, while the other one (Ben) tries to enforce them. We show a construction that is closer to the lower bound of 6 showed in [2] using entropy compression while building on the probabilistic arguments based on a version of the Lovász Local Lemma from [4]. Our result provides an explicit strategy for Ann to avoid (non-trivial) repetitions over a 7-letter alphabet.

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“…In the proof of Theorem 2, we will make use of a refined version of the Local Lemma introduced recently by Pegden [21] for the purpose of studying Thue games. A similar version was used by Peres and Schlag [22] in proving a Diophantine approximation result on lacunary sets of integers.…”

Section: Theorem 2 Every Path P N Satisfiesmentioning

confidence: 99%

“…We shall use the following version of the Local Lemma, which was proved in [21]. We state it in a restricted form which is sufficient for our purposes.…”

Section: The Local Lemmamentioning

confidence: 99%

Nonrepetitive list colourings of paths

Grytczuk

Przybyło

Zhu

2010

Random Struct Algorithms

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ABSTRACT:A vertex colouring c of a graph G is called nonrepetitive if for every integer r ≥ 1 and every path P = (v 1 , v 2 , . . . , v 2r ) in G, the first half of P is coloured differently from the second half of P, that is, c(v j ) = c(v r+j ) for some j = 1, 2, . . . , r. This notion was inspired by a striking result of Thue asserting that the path P n on n vertices has a nonrepetitive three-colouring, no matter how large n is. A k-list assignment of a graph G is a mapping L which assigns a set L(v) of k permissible colours to each vertex v of G. The Thue choice number of G, denoted by π ch (G), is the least integer k such that for every k-list assignment L there is a nonrepetitive colouring c of G satisfying c(v) ∈ L(v) for every vertex v of G. Using the Lefthanded Local Lemma we prove that π ch (P n ) ≤ 4 for every n.

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Highly nonrepetitive sequences: Winning strategies from the local lemma

Cited by 28 publications

References 22 publications

On the Facial Thue Choice Index via Entropy Compression

On the Facial Thue Choice Index via Entropy Compression

A note on Thue games

Nonrepetitive list colourings of paths

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