2018
DOI: 10.48550/arxiv.1806.06149
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A note on choosability with defect 1 of graphs on surfaces

Abstract: is note proves that every graph of Euler genus µ is ⌈2 + √ 3µ + 3⌉-choosable with defect 1 (that is, clustering 2). us, allowing defect as small as 1 reduces the choice number of surface embeddable graphs below the chromatic number of the surface. For example, the chromatic number of the family of toroidal graphs is known to be 7. e bound above implies that toroidal graphs are 5-choosable with defect 1. is strengthens the result of Cowen, Goddard and Jesurum (1997) who showed that toroidal graphs are 5-coloura… Show more

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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%
See 1 more Smart Citation
“…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%