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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.
CHARACTERISATIONS AND EXAMPLES OF GRAPH CLASSES WITH BOUNDED EXPANSION
"... Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of t ..."
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Cited by 9 (3 self)
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Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with Erdös-Rényi model of random graphs with constant average degree. In particular, we prove that for every fixed d> 0, there exists a class with bounded expansion, such that a random graph of order n and edge probability d/n asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded non-repetitive chromatic number. We also prove that graphs with ‘linear ’ crossing number are contained in a topologically-closed class, while graphs with bounded crossing number are contained in a minor-closed class.
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 tree-partition-width
, 2006
"... Abstract. A tree-partition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The tree-partition-width of G is the minimum number of vertices in a bag in a tree-partition of G. An anonymous referee of the paper by Ding ..."
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Cited by 6 (3 self)
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Abstract. A tree-partition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The tree-partition-width of G is the minimum number of vertices in a bag in a tree-partition of G. An anonymous referee of the paper by Ding and Oporowski [J. Graph Theory, 1995] proved that every graph with tree-width k ≥ 3 and maximum degree ∆ ≥ 1 has tree-partition-width at most 24k∆. We prove that this bound is within a constant factor of optimal. In particular, for all k ≥ 3 and for all sufficiently large ∆, we construct a graph with tree-width k, maximum degree ∆, and tree-partition-width at least ( 1 8 upper bound to 5
Characterizations and Examples of Graph Classes with bounded expansion
"... Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of th ..."
Abstract
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Cited by 2 (1 self)
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Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with Erdös-Rényi model of random graphs with constant average degree. In particular, we prove that for every fixed d >
September 24-28, 2012 Nonrepetitive, acyclic and clique colorings of graphs with few P 4 's
"... Abstract In this paper, we propose algorithms to determine the Thue chromatic number and the clique chromatic number of P 4 -tidy graphs and (q, q − 4)-graphs. These classes include cographs and P 4 -sparse graphs. All algorithms have linear-time complexity, for fixed q, and then are fixed paramete ..."
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Abstract In this paper, we propose algorithms to determine the Thue chromatic number and the clique chromatic number of P 4 -tidy graphs and (q, q − 4)-graphs. These classes include cographs and P 4 -sparse graphs. All algorithms have linear-time complexity, for fixed q, and then are fixed parameter tractable. All these coloring problems are known to be NP-hard for general graphs. 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, generalizing a result from
Nonrepetitive colorings of lexicographic product of paths and other graphs
, 2013
"... A coloring c of the vertices of a graph G is nonrepetitive if there exists no path v1v2... v2l for which c(vi) = c(vl+i) for all 1 ≤ i ≤ l. Given graphs G and H with |V (H) | = k, the lexicographic product G[H] is the graph obtained by substituting every vertex ofG by a copy ofH, and every edge of ..."
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A coloring c of the vertices of a graph G is nonrepetitive if there exists no path v1v2... v2l for which c(vi) = c(vl+i) for all 1 ≤ i ≤ l. Given graphs G and H with |V (H) | = k, the lexicographic product G[H] is the graph obtained by substituting every vertex ofG by a copy ofH, and every edge ofG by a copy ofKk,k. We prove that for a sufficiently long path P, a nonrepetitive coloring of P [Kk] needs at least 3k + bk/2c colors. If k> 2 then we need exactly 2k + 1 colors to nonrepetitively color P [Ek], where Ek is the empty graph on k vertices. If we further require that every copy of Ek be rainbow-colored and the path P is sufficiently long, then the smallest number of colors needed for P [Ek] is at least 3k + 1 and at most 3k + dk/2e. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results.
