Defective and Clustered Graph Colouring

Wood, David R.

doi:10.37236/7406

Cited by 44 publications

(51 citation statements)

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“…Note that Havet and Sereni [27] gave a construction to show that no lower value of k is possible. That is, for m ∈ R + , the defective chromatic number of the class of graphs with maximum average degree m equals m/2 + 1; see also [42].…”

Section: Defective Choosabilitymentioning

confidence: 99%

“…Now consider clustered colouring of earth-moon graphs. Wood [42] describes examples of earth-moon graphs that are not 5-colourable with bounded clustering. Thus the clustered chromatic number of earth-moon graphs is at least 6.…”

Section: Earth-moon Colouring and Thicknessmentioning

confidence: 99%

“…Now consider clustered colourings of graphs with given thickness. Obviously, the clustered chromatic number of graphs with thickness t is at most 6t, and Wood [42] proved a lower bound of 2t + 2. Since every graph with thickness t has maximum average degree strictly less than 6t, Theorems 1.4, 1.5 and 1.6 imply the following improved upper bounds.…”

Section: Theorem 82 ([40]mentioning

confidence: 99%

“…Since every graph with thickness t has maximum average degree strictly less than 6t, Theorems 1.4, 1.5 and 1.6 imply the following improved upper bounds. Thickness is generalized as follows; see [32,40,42]. For an integer g 0, the g-thickness of a graph G is the minimum integer t such that G is the union of t subgraphs each with Euler genus at most g. Ossona de Mendez, Oum and Wood [40] determined the defective chromatic number of this class as follows (thus generalizing Theorem 8.2).…”

Section: Theorem 82 ([40]mentioning

confidence: 99%

“…Defective and clustered (list-) colouring has been widely studied on a variety of graph classes, including bounded maximum degree [2,29], planar [14,16,23], bounded genus [3,13,14,15,25,43], excluding a minor [21,24,30,36,39,40], excluding a topological minor [21,40], and excluding an immersion [30]. See [42] for a survey on defective and clustered colouring. All of these classes have bounded maximum average degree.…”

Section: Introductionmentioning

confidence: 99%

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Defective and clustered choosability of sparse graphs

Hendrey

Wood

2019

Combinator. Probab. Comp.

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An (improper) graph colouring has defect d if each monochromatic subgraph has maximum degree at most d, and has clustering c if each monochromatic component has at most c vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than 2d+2 d+2 k is k-choosable with defect d. This improves upon a similar result by Havet and Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with maximum average degree m, no (1 − ǫ)m bound on the number of colours was previously known. The above result with d = 1 solves this problem. It implies that every graph with maximum average degree m is ⌊ 3 4 m + 1⌋-choosable with clustering 2. This extends a result of Kopreski and Yu [Discrete Math., 2017] to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degree m is ⌊ 7 10 m + 1⌋-choosable with clustering 9, and is ⌊ 2 3 m + 1⌋-choosable with clustering O(m). As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth-moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.

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Section: Defective Choosabilitymentioning

confidence: 99%

Section: Earth-moon Colouring and Thicknessmentioning

confidence: 99%

Section: Theorem 82 ([40]mentioning

confidence: 99%

Section: Theorem 82 ([40]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Defective and clustered choosability of sparse graphs

Hendrey

Wood

2019

Combinator. Probab. Comp.

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Decomposing planar graphs into graphs with degree restrictions

Cho

Choi

Kim

et al. 2022

Journal of Graph Theory

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Given a graph G $G$, a decomposition of G $G$ is a partition of its edges. A graph is ( d , h ) $(d,h)$‐decomposable if its edge set can be partitioned into a d $d$‐degenerate graph and a graph with maximum degree at most h $h$. For d ≤ 4 $d\le 4$, we are interested in the minimum integer h d ${h}_{d}$ such that every planar graph is ( d , h d ) $(d,{h}_{d})$‐decomposable. It was known that h 3 ≤ 4 ${h}_{3}\le 4$, h 2 ≤ 8 ${h}_{2}\le 8$, and h 1 = ∞ ${h}_{1}=\infty $. This paper proves that h 4 = 1 , h 3 = 2 ${h}_{4}=1,{h}_{3}=2$, and 4 ≤ h 2 ≤ 6 $4\le {h}_{2}\le 6$.

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Polynomial bounds for chromatic number. III. Excluding a double star

Scott

Seymour

Spirkl

2022

Journal of Graph Theory

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A “double star” is a tree with two internal vertices. It is known that the Gyárfás–Sumner conjecture holds for double stars, that is, for every double star H $H$, there is a function f H ${f}_{H}$ such that if G $G$ does not contain H $H$ as an induced subgraph then χ ( G ) ≤ f H ( ω ( G ) ) $\chi (G)\le {f}_{H}(\omega (G))$ (where χ , ω $\chi ,\omega $ are the chromatic number and the clique number of G $G$). Here we prove that f H ${f}_{H}$ can be chosen to be a polynomial.

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Defective and Clustered Graph Colouring

Cited by 44 publications

References 1 publication

Defective and clustered choosability of sparse graphs

Defective and clustered choosability of sparse graphs

Decomposing planar graphs into graphs with degree restrictions

Polynomial bounds for chromatic number. III. Excluding a double star

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