Nonrepetitive list colourings of paths

Grytczuk, Jarosław; Przybyło, Jakub; Zhu, Xuding

doi:10.1002/rsa.20347

Cited by 33 publications

(36 citation statements)

References 22 publications

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“…In fact, the bound 64 comes as a special case of a result on nonrepetitive colorings of bounded degree graphs (Alon et al [2]; see also [13]). Recently Grytczuk, Przyby lo and Zhu [15] proved that lists of sizes at least 4 suffice. They achieve this almost tight bound applying an enhanced version of the local lemma due to Pegden [21].…”

Section: Introductionmentioning

confidence: 99%

New approach to nonrepetitive sequences

Grytczuk

Kozik

Micek

2012

Random Struct Algorithms

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A sequence is nonrepetitive if it does not contain two adjacent identical blocks. The remarkable construction of Thue asserts that three symbols are enough to build an arbitrarily long nonrepetitive sequence. It is still not settled whether the following extension holds: for every sequence of three‐element sets L1,…,Ln there exists a nonrepetitive sequence s1,…,sn with si∈Li. We propose a new non‐constructive way to build long nonrepetitive sequences and provide an elementary proof that sets of size 4 suffice confirming the best known bound. The simple double counting in the heart of the argument is inspired by the recent algorithmic proof of the Lovász local lemma due to Moser and Tardos. Furthermore we apply this approach and present game‐theoretic type results on nonrepetitive sequences. Nonrepetitive game is played by two players who pick, one by one, consecutive terms of a sequence over a given set of symbols. The first player tries to avoid repetitions, while the second player, in contrast, wants to create them. Of course, by simple imitation, the second player can force lots of repetitions of size 1. However, as proved by Pegden, there is a strategy for the first player to build an arbitrarily long sequence over 37 symbols with no repetitions of size greater than 1. Our techniques allow to reduce 37–6. Another game we consider is the erase‐repetition game. Here, whenever a repetition occurs, the repeated block is immediately erased and the next player to move continues the play. We prove that there is a strategy for the first player to build an arbitrarily long nonrepetitive sequence over 8 symbols. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

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Section: Introductionmentioning

confidence: 99%

New approach to nonrepetitive sequences

Grytczuk

Kozik

Micek

2012

Random Struct Algorithms

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“…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). This result was recently reproved in the article of Grytczuk et al.…”

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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“…By means of a sophisticated application of a special version of the Local Lemma, it was shown in [16] that π ch (P n ) ≤ 4 for every path P n . It was a significant improvement of the previously known bounds, see [16] for details. (Whether the optimal bound in this list version of the original Thue problem is 3 or 4 is still an intriguing open problem).…”

Section: Families With Constant Upper Boundsmentioning

confidence: 99%

On the Facial Thue Choice Number of Plane Graphs Via Entropy Compression Method

Przybyło

Schreyer

Škrabuľáková

2015

Graphs and Combinatorics

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Let G be a plane graph. A vertex-colouring ϕ of G is called facial non-repetitive if for no sequence r 1 r 2 . . . r 2n , n ≥ 1, of consecutive vertex colours of any facial path it holds r i = r n+i for all i = 1, 2, . . . , n. A plane graph G is facial non-repetitively l-choosable if for every list assignment L : V → 2 N with minimum list size at least l there is a facial non-repetitive vertex-colouring ϕ with colours from the associated lists. The facial Thue choice number, π f l (G), of a plane graph G is the minimum number l such that G is facial non-repetitively l-choosable. We use the so-called entropy compression method to show that π f l (G) ≤ c∆ for some absolute constant c and G a plane graph with maximum degree ∆. Moreover, we give some better (constant) upper bounds on π f l (G) for special classes of plane graphs.

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Nonrepetitive list colourings of paths

Cited by 33 publications

References 22 publications

New approach to nonrepetitive sequences

New approach to nonrepetitive sequences

On the Facial Thue Choice Index via Entropy Compression

On the Facial Thue Choice Number of Plane Graphs Via Entropy Compression Method

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