Colouring graphs with bounded generalized colouring number

Zhu, Xuding

doi:10.1016/j.disc.2008.03.024

Cited by 99 publications

(96 citation statements)

References 12 publications

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“…• It follows from [26,Lemma 3.3] that for 3-regular graphs of high girth the weak r-colouring numbers grow exponentially with r. Hence the polynomial bound for wcol r (G) in Theorem 1.2 for classes with excluded minors cannot be extended to classes with bounded degree, or even to classes with excluded topological minors.…”

Section: Theorem 15mentioning

confidence: 99%

“…But maybe the most natural generalisation of the colouring numbers is the two series col r and wcol r of generalised colouring numbers introduced by Kierstead and Yang [16] in the context of colouring games and marking games on graphs. As proved by Zhu [26], these invariants are strongly related to low tree-depth decompositions [19], and can be used to characterise bounded expansion classes of graphs (introduced in [20]) and nowhere dense classes of graphs (introduced in [21]). For more details on this connection, we refer the interested reader to [22].…”

Section: Introductionmentioning

confidence: 99%

“…Using probabilistic arguments, Zhu [26] was the first to give a non-trivial bound for col r (G) in terms of the densities of shallow minors of G. For a graph G excluding a complete graph K t as a minor, Zhu's bound gives col r (G) ≤ 1 + q r , where q 1 is the maximum average degree of a minor of G, and q i is inductively defined by q i+1 = q 1 · q 2i 2 i . Grohe et al [10] improved Zhu's bounds as follows:…”

Section: Introductionmentioning

confidence: 99%

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On the generalised colouring numbers of graphs that exclude a fixed minor

Heuvel

Mendez

Rabinovich

et al. 2015

Electronic Notes in Discrete Mathematics

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The generalised colouring numbers col r (G) and wcol r (G) were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications.In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the r-colouring number col r and a polynomial bound for the weak r-colouring number wcol r . In particular, we show that if G excludes K t as a minor, for some fixed t ≥ 4, then col r (G) ≤ t−1 2 (2r + 1) and wcol r (G) ≤ r+t−2 t−2 · (t − 3)(2r + 1) ∈ O(r t−1 ). In the case of graphs G of bounded genus g, we improve the bounds to col r (G) ≤ (2g + 3)(2r + 1) (and even col r (G) ≤ 5r + 1 if g = 0, i.e. if G is planar) and wcol r (G) ≤ 2g + r+2 2 (2r + 1).

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Section: Theorem 15mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the generalised colouring numbers of graphs that exclude a fixed minor

Heuvel

Mendez

Rabinovich

et al. 2015

Electronic Notes in Discrete Mathematics

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“…In this paper we prove that any class with bounded expansion is quasi-wide (Theorem 3.19). Bounded expansion classes and algorithmic applications have been introduced in [NOdM05a,NOdM05b,NOdM06a,NOdM06b,NOdM07,NOdM08a,NOdM08b] and have been discussed in Z. Dvořák's PhD thesis [Dvo07a] and in X. Zhu paper [Zhu06].…”

Section: Supported By Grant 1m0021620808 Of the Czech Ministry Of Edumentioning

confidence: 99%

First order properties on nowhere dense structures

Nešetřil¹,

Mendez²

2010

J. symb. log.

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Abstract. A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed families). This in turn led to the notions of wide, almost wide and quasi-wide classes of graphs. It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativized) homomorphism preservation theorem holds for them. In this paper we show that (more general) classes with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) nowhere dense classes are quasi wide. This not only strictly generalizes the previous results but it also provides new proofs and algorithms for some of the old results. It appears that bounded expansion and nowhere dense classes are perhaps a proper setting for investigation of wide-type classes as in several instances we obtain a structural characterization. This also puts classes of bounded expansion in the new context. Our motivation stems from finite dualities. As a corollary we obtain that any homomorphism closed first order definable property restricted to a bounded expansion class is a restricted duality.

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“…According to [15] there exists for each p ∈ N a polynomial Q p such that every graph G has a vertex colouring c :…”

Section: Classes Of Graphs With Bounded Expansionmentioning

confidence: 99%

Colouring edges with many colours in cycles

Nešetřil¹,

Mendez²,

Zhu³

2014

Journal of Combinatorial Theory, Series B

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The arboricity of a graph G is the minimum number of colours needed to colour the edges of G so that every cycle gets at least two colours. Given a positive integer p, we define the generalized p-arboricity Arbp(G) of a graph G as the minimum number of colours needed to colour the edges of a multigraph G in such a way that every cycle C gets at least min(|C|, p + 1) colours. In the particular case where G has girth at least p + 1, Arbp(G) is the minimum size of a partition of the edge set of G such that the union of any p parts induce a forest. If we require further that the edge colouring be proper, i.e., adjacent edges receive distinct colours, then the minimum number of colours needed is the generalized p-acyclic edge chromatic number of G. In this paper, we relate the generalized p-acyclic edge chromatic numbers and the generalized p-arboricities of a graph G to the density of

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Colouring graphs with bounded generalized colouring number

Cited by 99 publications

References 12 publications

On the generalised colouring numbers of graphs that exclude a fixed minor

On the generalised colouring numbers of graphs that exclude a fixed minor

First order properties on nowhere dense structures

Colouring edges with many colours in cycles

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