A note on choosability with defect 1 of graphs on surfaces

Dujmović, Vida; Djedjiga, Outioua

doi:10.48550/arxiv.1806.06149

Cited by 1 publication

(2 citation statements)

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“…As an example of Theorem 4, it follows from Euler's formula that toroidal graphs have maximum average degree at most 6, implying every toroidal graph is 5-choosable with defect 1 and clustering 2, which was first proved by Dujmović and Outioua [18]. Previously, Cowen et al [14] proved that every toroidal graph is 5-colourable with defect 1.…”

Section: Theorem 3 ([34]mentioning

confidence: 93%

“…First consider defective colourings of earth-moon graphs. Since the maximum average degree of every earth-moon graph is less than 12, Theorem 1 by Havet and Sereni [27] implies that every earth-moon graph is k-choosable with defect d, for (k, d) ∈ { (7,18), (8,9), (9,5), (10,3), (11,2)}. This result gives no bound with at most 6 colours.…”

Section: Earth-moon Colouring and Thicknessmentioning

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: Theorem 3 ([34]mentioning

confidence: 93%