Thue type problems for graphs, points, and numbers

Grytczuk, Jarosław

doi:10.1016/j.disc.2007.08.039

Cited by 64 publications

(57 citation statements)

References 49 publications

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“…More than 100 years ago Thue [9] proved that there are nonrepetitive sequences of arbitrary length using only the symbols 1, 2, and 3. This result has been extended in many directions and a recent survey of Grytczuk [5] collects a substantial number of such results. In this paper we consider an extension to graph theory suggested by Alon et al [2].…”

Section: Introductionmentioning

confidence: 89%

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Nonrepetitive colorings of graphs of bounded tree-width

Kündgen

Pelsmajer

2008

Discrete Mathematics

112

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A sequence of the form s 1 s 2 . . . s m s 1 s 2 . . . s m is called a repetition. A vertex-coloring of a graph is called nonrepetitive if none of its paths is repetitively colored. We answer a question of Grytczuk [Thue type problems for graphs, points and numbers, manuscript] by proving that every outerplanar graph has a nonrepetitive 12-coloring. We also show that graphs of tree-width t have nonrepetitive 4 t -colorings.

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Section: Introductionmentioning

confidence: 89%

“…The answer to Question 1 is still open, but we provide some evidence suggesting an affirmative answer. At the Budapest workshop in honor of Miklós Simonovits' 60th birthday, Grytczuk suggested replacing planar by outerplanar as a first point of attack (see also [5]). …”

Section: Question 1 Is There a Constant K Such That Every Planar Gramentioning

confidence: 98%

Nonrepetitive colorings of graphs of bounded tree-width

Kündgen

Pelsmajer

2008

Discrete Mathematics

112

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“…The smallest k such that H has a non-repetitive k-coloring is called the Thue number π(H) of H [Thue 1906]. Alon et al [2002] showed via the LLL that π(H) ≤ O( (H) 2 ), where is the maximum degree of any vertex in H. This was followed by many additional works [Alon and Grytczuk 2008;Bresar et al 2007;Currie 2005;Grytczuk 2008;Kündgen and Pelsmajer 2008;Schaefer and Umans 2002]. However, no efficient construction is known till date, except for special classes of graphs such as complete graphs, cycles and trees.…”

Section: Applications That Avoid All Bad Eventsmentioning

confidence: 99%

New Constructive Aspects of the Lovász Local Lemma

Haeupler

Saha

Srinivasan

2011

J. ACM

123

190

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The Lovász Local Lemma (LLL) is a powerful tool that gives sufficient conditions for avoiding all of a given set of "bad" events, with positive probability. A series of results have provided algorithms to efficiently construct structures whose existence is non-constructively guaranteed by the LLL, culminating in the recent breakthrough of Moser and Tardos [2010] for the full asymmetric LLL. We show that the output distribution of the Moser-Tardos algorithm well-approximates the conditional LLL-distribution, the distribution obtained by conditioning on all bad events being avoided. We show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results.We also show that when a LLL application provides a small amount of slack, the number of resamplings of the Moser-Tardos algorithm is nearly linear in the number of underlying independent variables (not events!), and can thus be used to give efficient constructions in cases where the underlying proof applies the LLL to super-polynomially many events. Even in cases where finding a bad event that holds is computationally hard, we show that applying the algorithm to avoid a polynomial-sized "core" subset of bad events leads to a desired outcome with high probability. This is shown via a simple union bound over the probabilities of noncore events in the conditional LLL-distribution, and automatically leads to simple and efficient Monte-Carlo (and in most cases RNC) algorithms. We demonstrate this idea on several applications. We give the first constant-factor approximation algorithm for the Santa Claus problem by making a LLL-based proof of Feige constructive. We provide Monte Carlo algorithms for acyclic edge coloring, nonrepetitive graph colorings, and Ramsey-type graphs. In all these applications, the algorithm falls directly out of the non-constructive LLL-based proof. Our algorithms are very simple, often provide better bounds than previous algorithms, and are in several cases the first efficient algorithms known.As a second type of application we show that the properties of the conditional LLL-distribution can be used in cases beyond the critical dependency threshold of the LLL: avoiding all bad events is impossible in these cases. As the first (even nonconstructive) result of this kind, we show that by sampling a selected smaller core from the LLL-distribution, we can avoid a fraction of bad events that is higher than the expectation. MAX k-SAT is an illustrative example of this.

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“…Nonrepetitive colourings have recently been widely studied; see the survey [15]. A number of graph classes are known to have bounded nonrepetitive chromatic number.…”

Section: Nonrepetitive Graph Colouringsmentioning

confidence: 99%

Layered Separators for Queue Layouts, 3D Graph Drawing and Nonrepetitive Coloring

Dujmović

Morin

Wood

2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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Abstract-Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω( √ n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor-closed families. We study a special type of graph separator, called a layered separator, which may have linear size in n, but has bounded size with respect to a different measure, called the breadth. We prove that a wide class of graphs admit layered separators of bounded breadth, including graphs of bounded Euler genus. We use layered separators to prove O(log n) bounds for a number of problems where O( √ n) was a long standing previous best bound. This includes the nonrepetitive chromatic number and queue-number of graphs with bounded Euler genus. We extend these results to all proper minor-closed families, with a O(log n) bound on the nonrepetitive chromatic number, and a log O(1) n bound on the queue-number. Only for planar graphs were log O(1) n bounds previously known. Our results imply that every graph from a proper minor-closed class has a 3-dimensional grid drawing with n log O(1) n volume, whereas the previous best bound was O(n 3/2 ). Readers interested in the full details should consult arXiv:1302.0304 and arXiv:1306.1595, rather than the current extended abstract.

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Thue type problems for graphs, points, and numbers

Cited by 64 publications

References 49 publications

Nonrepetitive colorings of graphs of bounded tree-width

Nonrepetitive colorings of graphs of bounded tree-width

New Constructive Aspects of the Lovász Local Lemma

Layered Separators for Queue Layouts, 3D Graph Drawing and Nonrepetitive Coloring

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