Nonrepetitive colorings of graphs

Alon, Noga; Grytczuk, Jarosław

doi:10.1016/j.endm.2005.06.077

Cited by 23 publications

(46 citation statements)

References 31 publications

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“…This is a significant difference between σ and τ , as for instance τ (K 1,n ) = 2 and σ(K 1,n ) n + 1 for the stars K 1,n . In fact, τ (G) = O(∆ 2 ) was proved in [1]. As far as the authors know there is no similar upper bound for σ in terms of ∆.…”

Section: Introductionmentioning

confidence: 92%

“…Alon et al [1] asked whether τ is bounded by an absolute constant for planar graphs. After long considerations we believe the answer is affirmative, so we formulate it here in a positive statement, and give some evidences below.…”

Section: Outerplanar and Planar Graphsmentioning

confidence: 99%

“…Alon at al. [1] proved that τ (T ) 4 for any tree T . They asked whether a similar constant upper bound can be given for the class of planar graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Recently several generalizations to graphs have been suggested in [1] and [4]. The notions and names are still flexible, so we do not follow our predecessors completely.…”

Section: Introductionmentioning

confidence: 99%

“…We recall a sequence from [1] avoiding both squares and palindromes. We start with a square-free sequence, for instance with Thue's construction.…”

Section: Introductionmentioning

confidence: 99%

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On square-free vertex colorings of graphs

Barát

Varjú

2007

Studia Scientiarum Mathematicarum Hungarica

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A sequence of symbols a1, a2 . . . is called square-free if it does not contain a subsequence of consecutive terms of the form x1, . . . , xm, x1, . . . , xm. A century ago Thue showed that there exist arbitrarily long square-free sequences using only three symbols. Sequences can be thought of as colors on the vertices of a path. Following the paper of Alon, Grytczuk, Ha luszczak and Riordan, we examine graph colorings for which the color sequence is square-free on any path. The main result is that the vertices of any k-tree have a coloring of this kind using O(c k ) colors if c > 6. Alon et al. conjectured that a fixed number of colors suffices for any planar graph. We support this conjecture by showing that this number is at most 12 for outerplanar graphs. On the other hand we prove that some outerplanar graphs require at least 7 colors. Using this latter we construct planar graphs, for which at least 10 colors are necessary.

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Section: Introductionmentioning

confidence: 92%

Section: Outerplanar and Planar Graphsmentioning

confidence: 99%

“…Alon at al. [1] proved that τ (T ) 4 for any tree T . They asked whether a similar constant upper bound can be given for the class of planar graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Recently several generalizations to graphs have been suggested in [1] and [4]. The notions and names are still flexible, so we do not follow our predecessors completely.…”

Section: Introductionmentioning

confidence: 99%

“…We recall a sequence from [1] avoiding both squares and palindromes. We start with a square-free sequence, for instance with Thue's construction.…”

Section: Introductionmentioning

confidence: 99%

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On square-free vertex colorings of graphs

Barát

Varjú

2007

Studia Scientiarum Mathematicarum Hungarica

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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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Two Fixed-Parameter Algorithms for the Cocoloring Problem

Campos

Klein

Sampaio

et al. 2011

Algorithms and Computation

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In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number, the Thue chromatic number, the harmonious chromatic number and the clique chromatic number of P4-tidy graphs and (q, q−4)-graphs, for every fixed q. These classes include cographs, P4-sparse and P4-lite graphs. All these coloring problems are known to be NP-hard for general graphs. These algorithms are fixed parameter tractable on the parameter q(G), which is the minimum q such that G is a (q, q−4)-graph. We also prove that every connected (q, q − 4)-graph with at least q vertices is 2-clique-colorable and that every acyclic coloring of a cograph is also nonrepetitive.

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Nonrepetitive colorings of graphs

Cited by 23 publications

References 31 publications

On square-free vertex colorings of graphs

On square-free vertex colorings of graphs

New approach to nonrepetitive sequences

Two Fixed-Parameter Algorithms for the Cocoloring Problem

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