Planar graphs are 1-relaxed, 4-choosable

Cushing, William; Kierstead, H. A.

doi:10.1016/j.ejc.2009.11.013

Cited by 47 publications

(43 citation statements)

References 4 publications

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“…Label the vertices of U as b 1 , b 2 , b 3 in such a way that b 3 does not have the same color as either b 1 or b 2 , which is possible since the colors of these three vertices are not all equal. Then the result is exactly Theorem 2(A) of [4].…”

Section: (4 1) * -Coloringsmentioning

confidence: 73%

Defective choosability of graphs in surfaces

Woodall

2011

Discuss. Math. Graph Theory

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It is known that if G is a graph that can be drawn without edges crossing in a surface with Euler characteristic ǫ, and k and d are positive integers such that k 3 and d is sufficiently large in terms of k and ǫ, then G is (k, d) *-colorable; that is, the vertices of G can be colored with k colors so that each vertex has at most d neighbors with the same color as itself. In this paper, the known lower bound on d that suffices for this is reduced, and an analogous result is proved for list colorings (choosability). Also, the recent result of Cushing and Kierstead, that every planar graph is (4, 1) *-choosable, is extended to K 3,3-minor-free and K 5-minor-free graphs.

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Section: (4 1) * -Coloringsmentioning

confidence: 73%

Defective choosability of graphs in surfaces

Woodall

2011

Discuss. Math. Graph Theory

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“…For the rest of the proof we assume that every vertex x in layers L 1 , L 2 of D(l, k) has at most 14 bad neighbors in P j (x), j ∈ [2]. Let x be a good vertex in L 2 (such a vertex x exists as k > 14) and let y be any neighbor of x in L 2 .…”

Section: Lister's Strategymentioning

confidence: 99%

Defective 3-Paintability of Planar Graphs

Gutowski

Han

Krawczyk

et al. 2018

Electron. J. Combin.

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A d-defective k-painting game on a graph G is played by two players: Lister and Painter. Initially, each vertex is uncolored and has k tokens. In each round, Lister marks a chosen set M of uncolored vertices and removes one token from each marked vertex. In response, Painter colors vertices in a subset X of M which induce a subgraph G[X] of maximum degree at most d. Lister wins the game if at the end of some round there is an uncolored vertex that has no more tokens left. Otherwise, all vertices eventually get colored and Painter wins the game. We say that G is ddefective k-paintable if Painter has a winning strategy in this game. In this paper we show that every planar graph is 3-defective 3-paintable and give a construction of a planar graph that is not 2-defective 3-paintable.

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“…They both asked the question whether every planar graph is 1-defective 4-choosable. One decade later, Cushing and Kierstead [4] answered this question in the affirmative.…”

Section: Introductionmentioning

confidence: 99%

Locally planar graphs are 2-defective 4-paintable

Han

Zhu

2016

European Journal of Combinatorics

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Assume L is a k-list assignment of a graph G. A d-defective m-fold L-colouring φ of G assigns to each vertex v a set φ(v) of m colours, so that φ(v) ⊆ L(v) for each vertex v, and for each colour i, the set {v : i ∈ φ(v)} induces a subgraph of maximum degree at most d. In this paper, we consider on-line list d-defective m-fold colouring of graphs, where the list assignment L is given on-line, and the colouring is constructed on-line. To be precise, the d-defective (k, m)-painting game on a graph G is played by two players: Lister and Painter. Initially, each vertex has k tokens and is uncoloured. In each round, Lister chooses a set M of vertices and removes one token from each chosen vertex. Painter colours a subset X of M which induces a subgraph G[X] of maximum degree at most d. A vertex v is fully coloured if v has received m colours. Lister wins if at the end of some round, there is a vertex with no more tokens left and is not fully coloured. Otherwise, at some round, all vertices are fully coloured and Painter wins. We say G is d-defective (k, m)-paintable if Painter has a winning strategy in this game. This paper proves that every planar graph is 1-defective (9, 2)-paintable.

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Planar graphs are 1-relaxed, 4-choosable

Cited by 47 publications

References 4 publications

Defective choosability of graphs in surfaces

Defective choosability of graphs in surfaces

Defective 3-Paintability of Planar Graphs

Locally planar graphs are 2-defective 4-paintable

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