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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 kchoosable 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 kchoosable 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 ∆ 2choosable, 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 5choosable. The proofs of both these results are based on the MoserTardos entropycompression 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.
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 kcolouring. 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 kcolouring. 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 nvertex planar graphs. We prove a O(log n) upper bound. 1
FURTHER APPLICATIONS OF A POWER SERIES METHOD FOR PATTERN AVOIDANCE
, 2009
"... In combinatorics on words, a word w over an alphabet Σ is said to avoid a pattern p over an alphabet ∆ if there is no factor x of w and no nonerasing morphism h from ∆ ∗ to Σ ∗ such that h(p) = x. Bell and Goh have recently applied an algebraic technique due to Golod to show that for a certain ..."
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Cited by 3 (0 self)
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In combinatorics on words, a word w over an alphabet Σ is said to avoid a pattern p over an alphabet ∆ if there is no factor x of w and no nonerasing morphism h from ∆ ∗ to Σ ∗ such that h(p) = x. Bell and Goh have recently applied an algebraic technique due to Golod to show that for a certain wide class of patterns p there are exponentially many words of length n over a 4letter alphabet that avoid p. We consider some further consequences of their work. In particular, we show that any pattern with k variables of length at least 4 k is avoidable on the binary alphabet. This improves an earlier bound due to Cassaigne and Roth.
THUE CHOOSABILITY OF TREES
"... A vertex colouring of a graph G is nonrepetitive if for any path P = (v1, v2,..., v2r) in G, the first half is coloured differently from the second half. The Thue choice number of G is the least integer ` such that for every `list assignment L of G, there exists a nonrepetitive Lcolouring of G. ..."
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Cited by 3 (1 self)
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A vertex colouring of a graph G is nonrepetitive if for any path P = (v1, v2,..., v2r) in G, the first half is coloured differently from the second half. The Thue choice number of G is the least integer ` such that for every `list assignment L of G, there exists a nonrepetitive Lcolouring of G. We prove that for any positive integer `, there is a tree T with pich(T)> `. On the other hand, it is proved that if G ′ is a graph of maximum degree ∆, and G is obtained from G ′ by attaching to each vertex v of G ′ a connected graph of treedepth at most z rooted at v, then pich(G) ≤ c(∆, z) for some constant c(∆, d) depending only on ∆ and z.
The Lefthanded Local Lemma characterizes chordal dependency graphs
, 2012
"... Shearer gave a general theorem characterizing the family L of dependency graphs labeled with probabilities pv which have the property that for any family of events with a dependency graph from L (whose vertexlabels are upper bounds on the probabilities of the events), there is a positive probabili ..."
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Shearer gave a general theorem characterizing the family L of dependency graphs labeled with probabilities pv which have the property that for any family of events with a dependency graph from L (whose vertexlabels are upper bounds on the probabilities of the events), there is a positive probability that none of the events from the family occur. We show that, unlike the standard Lovász Local Lemma—which is less powerful than Shearer’s condition on every nonempty graph—a recently proved ‘Lefthanded ’ version of the Local Lemma is equivalent to Shearer’s condition for all chordal graphs. This also leads to a simple and efficient algorithm to check whether a given labeled chordal graph is in L. 1