Defective choosability of graphs in surfaces

Woodall, Douglas R.

doi:10.7151/dmgt.1557

Cited by 19 publications

(16 citation statements)

References 14 publications

(8 reference statements)

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“…Woodall [228] improved Theorem 11 to show that every graph with Euler genus g is 3-choosable with defect max{9, 2 + √ 4g + 6}. Thus…”

Section: Defective Colouring Of Graphs On Surfacesmentioning

confidence: 98%

Defective and Clustered Graph Colouring

Wood

2018

Electron. J. Combin.

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Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has defect d if each monochromatic component has maximum degree at most d. A colouring has clustering c if each monochromatic component has at most c vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdière parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack-or queue-number, graphs excluding K t as a minor, graphs excluding K s,t as a minor, and graphs excluding an arbitrary graph H as a minor.

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“…Woodall [228] improved Theorem 11 to show that every graph with Euler genus g is 3-choosable with defect max{9, 2 + √ 4g + 6}. Thus…”

Section: Defective Colouring Of Graphs On Surfacesmentioning

confidence: 98%

Defective and Clustered Graph Colouring

Wood

2018

Electron. J. Combin.

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“…This was proved by Archdeacon , who showed that every graph of Euler genus

g

(c_{3}, c_{3}, c_{3})

‐colorable with

c_{3} = \max {15, false(1 / 2 false) (3 g - 8)}

. The value

c_{3}

was subsequently improved to

\max {12, 6 + \sqrt{6 g}}

by Cowen, Goddard, and Jesurum , and eventually to

\max {9, 2 + \sqrt{4 g + 6}}

by Woodall .…”

Section: Introductionmentioning

confidence: 94%

“…Note however that if we let the maximum degree of the second color class be a function of

g

, then the maximum degree of the third color class can be made sublinear: it can be derived from the main result of Woodall that every graph of Euler genus

g

(9, O (\sqrt{g}), O (\sqrt{g}))

‐colorable. In the next subsection, we will prove that every graph of Euler genus

g

(2, O (\sqrt{g}), O (\sqrt{g}))

‐colorable and the constant 2 there is best possible.…”

Section: Graphs On Surfacesmentioning

confidence: 99%

Improper coloring of graphs on surfaces

Choi

2018

Journal of Graph Theory

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A graph G is (d1,…,dk)‐colorable if its vertex set can be partitioned into k sets V1,-0.28em…,Vk, such that for each i∈{1,…,k}, the subgraph of G induced by Vi has maximum degree at most di. The Four Color Theorem states that every planar graph is (0,0,0,0)‐colorable, and a classical result of Cowen, Cowen, and Woodall shows that every planar graph is (2,2,2)‐colorable. In this paper, we extend both of these results to graphs on surfaces. Namely, we show that every graph embeddable on a surface of Euler genus g>0 is (0,0,0,9g−4)‐colorable and (2,2,9g−4)‐colorable. Moreover, these graphs are also (0,0,O(g),O(g))‐colorable and (2,O(g),O(g))‐colorable. We also prove that every triangle‐free graph that is embeddable on a surface of Euler genus g is (0,0,O(g))‐colorable. This is an extension of Grötzsch's Theorem, which states that triangle‐free planar graphs are (0,0,0)‐colorable. Finally, we prove that every graph of girth at least 7 that is embeddable on a surface of Euler genus g is (0,O(g))‐colorable. All these results are best possible in several ways as the girth condition is sharp, the constant maximum degrees cannot be improved, and the bounds on the maximum degrees depending on g are tight up to a constant multiplicative factor.

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“…Let us also mention that defective and fragmented colorings have also been considered for nonplanar graphs of bounded maximum degree , bounded number of vertices , and for minor‐free graphs . In natural generalizations, one allows different color classes to have different defect (see, e.g., ), or considers list‐coloring, which, in fact, is the case in many of the results above.…”

Section: Improper Colorings Of Planar Graphsmentioning

confidence: 99%

Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths

2016

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Recently, Borodin, Kostochka, and Yancey (On 1-improper 2-coloring of sparse graphs. Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least 7 can be 2-colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least 6 admits a vertex coloring in 2 colors such that each monochromatic component is a path of length at most 14. Moreover, we show a list version of this result. On the other hand, for each positive integer t ≥ 3, we construct a planar graph of girth 4 such that in any coloring of vertices in 2 colors there is a monochromatic path of length at least t. It remains open whether each planar graph of girth 5 admits a 2-coloring with no long monochromatic paths.

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Defective choosability of graphs in surfaces

Cited by 19 publications

References 14 publications

Defective and Clustered Graph Colouring

Defective and Clustered Graph Colouring

Improper coloring of graphs on surfaces

Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths

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