Nonrepetitive Colorings of GraphsA Survey

Grytczuk, Jarosław

doi:10.1155/2007/74639

Cited by 53 publications

(83 citation statements)

References 23 publications

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“…In [4], it was shown via the LLL that for any graph G with maximum degree ∆, π(G) = O(∆ 2 ). The original constant term in that paper was not tight; a variety of further papers such as [17,18,20] have brought it down further. The best currently-known bound is that π(G) ≤ (1 + o(1))∆ 2 [14].…”

Section: Thus the Expected Running Time Of Mt Ismentioning

confidence: 99%

Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

Harris

Srinivasan

2017

ACM Trans. Algorithms

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Moser & Tardos have developed a powerful algorithmic approach (henceforth "MT") to the Lovász Local Lemma (LLL); the basic operation done in MT and its variants is a search for "bad" events in a current configuration. In the initial stage of MT, the variables are set independently. We examine the distributions on these variables which arise during intermediate stages of MT. We show that these configurations have a more or less "random" form, building further on the "MT-distribution" concept of Haeupler et al. in understanding the (intermediate and) output distribution of MT. This has a variety of algorithmic applications; the most important is that bad events can be found relatively quickly, improving upon MT across the complexity spectrum: it makes some polynomial-time algorithms sub-linear (e.g., for Latin transversals, which are of basic combinatorial interest), gives lower-degree polynomial run-times in some settings, transforms certain super-polynomial-time algorithms into polynomial-time ones, and leads to Las Vegas algorithms for some coloring problems for which only Monte Carlo algorithms were known.We show that in certain conditions when the LLL condition is violated, a variant of the MT algorithm can still produce a distribution which avoids most of the bad events. We show in some cases this MT variant can run faster than the original MT algorithm itself, and develop the first-known criterion for the case of the asymmetric LLL. This can be used to find partial Latin transversals -improving upon earlier bounds of Stein (1975) -among other applications. We furthermore give applications in enumeration, showing that most applications (where we aim for all or most of the bad events to be avoided) have large solution sets. We do this by showing that the MT-distribution has large Rényi entropy.

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Section: Thus the Expected Running Time Of Mt Ismentioning

confidence: 99%

Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

Harris

Srinivasan

2017

ACM Trans. Algorithms

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“…The method of Thue does not seem to work in this case, and it was not clear for a while whether π(G) is bounded even for cubic graphs. However in [3] (see also [15]) it was proved that π(G) ≤…”

Section: Theorem 1 (Thue) π(P Nmentioning

confidence: 99%

“…It was conjectured in [15] that π ch (P n ) = 3 for every n ≥ 4. While the conjecture remains open, Theorem 2 narrows down the gap to 1.…”

Section: Theorem 2 Every Path P N Satisfiesmentioning

confidence: 99%

“…However, it is a bit surprising that this happens already for trees: it was recently proved in [14] that π ch (G) is unbounded in the class of trees, while π(T ) ≤ 4 for every tree T [15]. Perhaps one may push the number π ch (G) below some absolute constant by taking a sufficiently large subdivision of G, that is, replacing the edges of G with sufficiently long paths.…”

Section: Final Remarksmentioning

confidence: 99%

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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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“…The difference π mch (G)−π m (G) can be arbitrarily large. It is known [9,11] that for any tree T , π 2 (T ) ≤ 4, however, for any k, there is a tree T with π 2ch (T ) > k. For paths, it was shown in [12] that π 2ch (P n ) ≤ 4 for all n. However, whether or not π 2ch (P n ) = π 2 (P n ) for all n remains an open problem.…”

mentioning

confidence: 99%

$$(2+\epsilon )$$ ( 2 + ϵ ) -Nonrepetitive List Colouring of Paths

Zhao

Zhu

2015

Graphs and Combinatorics

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A sequence a 1 a 2 . . . a p is an r -repetition (for a real number r > 1) if p = rq for some positive integer q, and a j = a j+q for j = 1, 2, . . . , p − q. In other words, the sequence can be divided into r blocks where all the blocks are the same, say, all the blocks equal to a 1 a 2 . . . a q for some q ≥ 1, except that when r is not an integer, the last block is the prefix of a 1 ...a q of length (r − r )q . A colouring of the vertices of a graph G is r -nonrepetitive if there is no path in G for which the colour sequence of its vertices forms an r -repetition. The r -nonrepetitive chromatic number π r (G) of G is the minimum number of colours needed in an r -nonrepetitive colouring of G. 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 r -nonrepetitive choice number π rch (G) of G is the least integer k such that for every k-list assignment L, there is an r -nonrepetitive colouring c of G satisfying c(v) ∈ L(v) for every vertex v of G. A classical result of Thue asserts that π 2 (P n ) ≤ 3 for all n. It is known that π 2ch (P n ) ≤ 4 for all n. However, it remains an open problem whether π 2ch (P n ) ≤ 3 for all n. This paper proves that for any > 0, π (2+ )ch (P n ) ≤ 3 for all n.

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Nonrepetitive Colorings of GraphsA Survey

Abstract: A vertex coloringfof a graphGisnonrepetitiveif there are no integerr≥1and a simple pathv1,…,v2rinGsuch thatf(vi)=f(vr+i)for alli=1,…,r. This notion is a graph-theoretic variant of nonrepetitive sequences of Thue. The paper surveys problems and results on this topic.

Cited by 53 publications

References 23 publications

Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

Nonrepetitive list colourings of paths

$$(2+\epsilon )$$ ( 2 + ϵ ) -Nonrepetitive List Colouring of Paths

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Nonrepetitive Colorings of GraphsA Survey

Abstract: A vertex coloringfof a graphGisnonrepetitiveif there are no integerr≥1and a simple pathv1,…,v2rinGsuch thatf(vi)=f(vr+i)for alli=1,…,r. This notion is a graph-theoretic variant of nonrepetitive sequences of Thue. The paper surveys problems and results on this topic.

Cited by 53 publications

References 23 publications

Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

Algorithmic and Enumerative Aspects of the Moser-Tardos Distribution

Nonrepetitive list colourings of paths

$$(2+\epsilon )$$ ( 2 + ϵ ) -Nonrepetitive List Colouring of Paths

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Nonrepetitive Colorings of GraphsA Survey