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A note on treepartitionwidth
, 2006
"... Abstract. A treepartition 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 treepartitionwidth of G is the minimum number of vertices in a bag in a treepartition of G. An anonymous referee of the paper by Ding ..."
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Abstract. A treepartition 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 treepartitionwidth of G is the minimum number of vertices in a bag in a treepartition of G. An anonymous referee of the paper by Ding and Oporowski [J. Graph Theory, 1995] proved that every graph with treewidth k ≥ 3 and maximum degree ∆ ≥ 1 has treepartitionwidth 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 treewidth k, maximum degree ∆, and treepartitionwidth at least ( 1 8 upper bound to 5
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 5 (2 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 lineartime 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 socalled 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 ErdösRényi 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 nonrepetitive chromatic number. We also prove that graphs with ‘linear ’ crossing number are contained in a topologicallyclosed class, while graphs with bounded crossing number are contained in a minorclosed class.
Nonrepetitive Colouring via Entropy Compression
, 1112
"... 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 4 (3 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. 1
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 ..."
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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 lineartime 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 socalled 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 ErdösRényi model of random graphs with constant average degree. In particular, we prove that for every fixed d >
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 graphtheoretic variant of nonrepetitive sequences of Thue. The paper surveys problems and results on this topic. ..."
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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 graphtheoretic variant of nonrepetitive sequences of Thue. The paper surveys problems and results on this topic.
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 rainbowcolored 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.