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Nonrepetitive Colouring via Entropy Compression
, 2012
"... A vertex colouring of a graph is nonrepetitive if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively k-choosable if given lists of at least k colours at each vertex, there is a nonrepetitive colouringsuch that eachvertex iscolouredf ..."
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Cited by 12 (2 self)
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A vertex colouring of a graph is nonrepetitive if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively k-choosable if given lists of at least k colours at each vertex, there is a nonrepetitive colouringsuch that eachvertex iscolouredfrom its own list. It is knownthat everygraph with maximum degree∆is c ∆ 2-choosable, forsomeconstantc. We provethis result with c = 1 (ignoring lower order terms). We then prove that every subdivision of a graph with sufficiently many division vertices per edge is nonrepetitively 5-choosable. The proofs of both these results are based on the Moser-Tardos entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek for the nonrepetitive choosability of paths. Finally, we prove that every graph with pathwidth k is nonrepetitively O(k 2)-colourable.
Highly nonrepetitive sequences: winning strategies from the Local Lemma
, 2009
"... We prove game-theoretic versions of several classical results on nonrepetitive sequences, showing the existence of winning strategies using an extension of the Local Lemma which can dramatically reduce the number of edges needed in a dependency graph when there is an ordering underlying the signific ..."
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Cited by 11 (1 self)
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We prove game-theoretic versions of several classical results on nonrepetitive sequences, showing the existence of winning strategies using an extension of the Local Lemma which can dramatically reduce the number of edges needed in a dependency graph when there is an ordering underlying the significant dependencies of events. This appears to represent the first successful application of a Local Lemma to games.
Notes on nonrepetitive graph colouring
- ELECTRON. J. COMBIN
, 2008
"... A vertex colouring of a graph is nonrepetitive on paths if there is no path v1, v2,..., v2t such that vi and vt+i receive the same colour for all i = 1, 2,..., t. We determine the maximum density of a graph that admits a k-colouring that is nonrepetitive on paths. We prove that every graph has a sub ..."
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Cited by 9 (5 self)
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A vertex colouring of a graph is nonrepetitive on paths if there is no path v1, v2,..., v2t such that vi and vt+i receive the same colour for all i = 1, 2,..., t. We determine the maximum density of a graph that admits a k-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4-colouring that is nonrepetitive on paths. The best previous bound was 5. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree ∆ has a f(∆)-colouring that is nonrepetitive on walks. We prove that every graph with treewidth k and maximum degree ∆ has a O(k∆)-colouring that is nonrepetitive on paths, and a O(k ∆ 3)-colouring that is nonrepetitive on walks.
Nonrepetitive Colourings of Planar Graphs with O(log n) Colours
, 2012
"... A vertex colouring of a graph is nonrepetitive if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The nonrepetitive chromatic number of a graph G is the minimum integer k such that G has a nonrepetitive k-colouring. Whether planar gr ..."
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Cited by 6 (3 self)
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A vertex colouring of a graph is nonrepetitive if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The nonrepetitive chromatic number of a graph G is the minimum integer k such that G has a nonrepetitive k-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is O ( √ n) for n-vertex planar graphs. We prove a O(log n) upper bound. 1
Nonrepetitive Colorings of Graphs -- A Survey
, 2007
"... A vertex coloring f of a graph G is nonrepetitive if there are no integer r ≥ 1 and a simple path v1,...,v2r in G such that f (vi) = f (vr+i) for all i = 1,...,r. This notion is a graph-theoretic variant of nonrepetitive sequences of Thue. The paper surveys problems and results on this topic. ..."
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Cited by 6 (0 self)
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A vertex coloring f of a graph G is nonrepetitive if there are no integer r ≥ 1 and a simple path v1,...,v2r in G such that f (vi) = f (vr+i) for all i = 1,...,r. This notion is a graph-theoretic variant of nonrepetitive sequences of Thue. The paper surveys problems and results on this topic.
A note on non-repetitive colourings of planar graphs, arXiv:math/0307365 v 1
, 2003
"... Alon et al. introduced the concept of non-repetitive colourings of graphs. Here we address some questions regarding non-repetitive colourings of planar graphs. Specifically, we show that the faces of any outerplanar map can be non-repetitively coloured using at most five colours. We also give some l ..."
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Alon et al. introduced the concept of non-repetitive colourings of graphs. Here we address some questions regarding non-repetitive colourings of planar graphs. Specifically, we show that the faces of any outerplanar map can be non-repetitively coloured using at most five colours. We also give some lower bounds for the number of colours required to non-repetitively colour the vertices of both outerplanar and planar graphs. 1
Nonrepetitive sequences on arithmetic progressions
"... A sequence S = s1s2...sn is said to be nonrepetitive if no two adjacent blocks of S are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3-element set of symbols. We study a generalization of nonrepetitive sequences involving arithmetic progressions. We p ..."
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A sequence S = s1s2...sn is said to be nonrepetitive if no two adjacent blocks of S are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3-element set of symbols. We study a generalization of nonrepetitive sequences involving arithmetic progressions. We prove that for every k � 1, there exist arbitrarily long sequences over at most 2k +10 √ k symbols whose subsequences, indexed by arithmetic progressions with common differences from the set {1,2,...,k}, are nonrepetitive. This improves a previous bound of e 33 k obtained by
Infinite Sequences and Pattern Avoidance
, 2004
"... The study of combinatorics on words dates back at least to the beginning of the 20 century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains no squares (identical adjacent blocks) xx. This result was eventually used to so ..."
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The study of combinatorics on words dates back at least to the beginning of the 20 century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains no squares (identical adjacent blocks) xx. This result was eventually used to solve some longstanding open problems in algebra and has remarkable connections to other areas of mathematics and computer science as well. In this thesis we primarily study several variations of the problems studied by Thue in his work on repetitions in words, including some recent connections to other areas, such as graph theory.
Square-free colorings of graphs
"... Let G be a graph and let c be a coloring of its edges. If the se-quence of colors along a walk of G is of the form a1,..., an, a1,..., an, the walk is called a square walk. We say that the coloring c is square-free if any open walk is not a square and call the minimum number of colors needed so that ..."
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Let G be a graph and let c be a coloring of its edges. If the se-quence of colors along a walk of G is of the form a1,..., an, a1,..., an, the walk is called a square walk. We say that the coloring c is square-free if any open walk is not a square and call the minimum number of colors needed so that G has a square-free coloring a walk Thue num-ber and denote it by piw(G). This concept is a variation of the Thue number introduced by Alon, Grytczuk, Ha luszczak, and Riordan in [1]. Using the walk Thue number several results of [1] are extended. The Thue number of some complete graphs is extended to Hamming graphs. This result (for the case of hypercubes) is used to show that if a graph G on n vertices and m edges is the subdivision graph of some graph, then piw(G) ≤ n− m 2. Graph products are also considered. An inequality for the Thue number of the Cartesian product of trees is extended to arbitrary graphs and upper bounds for the (walk) Thue number of the direct and the strong products are also given. Using the latter results the (walk) Thue number of complete multipartite graphs is bounded which in turn gives a bound for arbitrary graphs in general and for perfect graphs in particular.
