Locally planar graphs are 5-choosable

DeVos, Matt; Kawarabayashi, Ken-ichi; Mohar, Bojan

doi:10.1016/j.jctb.2008.01.003

Cited by 30 publications

(35 citation statements)

References 16 publications

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“…However, only very few graphs have choosability close to this bound. DeVoss et al [2] obtained more general result claiming that locally planar graphs are 5-choosable, which extended the result of Thomassen to non-planar graphs. Furthermore, Kawarabayashi and Mohar [5] proved that there are only finitely many minimal graphs that are not 5-choosable on any fixed surface.…”

Section: Introductionmentioning

confidence: 74%

Graphs with Two Crossings Are 5-Choosable

Dvořák¹,

Lidický²,

Škrekovski³

2011

SIAM J. Discrete Math.

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Section: Introductionmentioning

confidence: 74%

Graphs with Two Crossings Are 5-Choosable

Dvořák¹,

Lidický²,

Škrekovski³

2011

SIAM J. Discrete Math.

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“…As we said, the proof of Theorem 3.1 is almost identical to that in [8], but let us give some intuition. The assumption of Theorem 3.1 implies that after deleting all the vertices in S 1 ∪ S 2 ∪ .…”

Section: List-coloring Extensions In Bounded-genus Graphsmentioning

confidence: 94%

“…Can we extend this precoloring to an L-coloring of the whole graph? To answer this question, we use the following tool developed by DeVos, Kawarabayashi and Mohar [8], which generalizes the graph-coloring case by Thomassen [33] to the list-coloring case. In fact, our statement below is different from the original, but it follows from the same proof as in [8] by combining with the RobertsonSeymour metric.…”

Section: List-coloring Extensions In Bounded-genus Graphsmentioning

confidence: 99%

“…And every vertex not on these cuffs has a list with at least six available colors. The result in [8] says that the resulting graph is even 5-list-colorable. But if we allow every vertex not on the cuffs to have a list with at least six available colors, then the proof becomes much easier even if all the vertices on the cuffs have a list with only four available colors.…”

Section: List-coloring Extensions In Bounded-genus Graphsmentioning

confidence: 99%

“…So the first condition implies that, for each bag, no matter how we color the apex vertices, each vertex in the surface part still has six available colors in its list (because we list-color the graph using at most χ l (G) + k − 2 colors). This allows us to use the result by DeVos, Kawarabayashi, and Mohar [8]. Specifically, the following is possible.…”

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confidence: 99%

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Additive Approximation Algorithms for List-Coloring Minor-Closed Class of Graphs

Kawarabayashi¹,

Demaine²,

Hajiaghayi³

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

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It is known that computing the list chromatic number is harder than computing the chromatic number (assuming NP = coNP). In fact, the problem of deciding whether a given graph is f -list-colorable for a function f : V → {c − 1, c} for c ≥ 3 is Π p 2 -complete. In general, it is believed that approximating list coloring is hard for dense graphs.In this paper, we are interested in sparse graphs. More specifically, we deal with nontrivial minor-closed classes of graphs, i.e., graphs excluding some K k minor. We refine the seminal structure theorem of Robertson and Seymour, and then give an additive approximation for list-coloring within k − 2 of the list chromatic number. This improves the previous multiplicative O(k)-approximation algorithm [20]. Clearly our result also yields an additive approximation algorithm for graph coloring in a minorclosed graph class. This result may give better graph colorings than the previous multiplicative 2-approximation algorithm for graph coloring in a minor-closed graph class [6].Our structure theorem is of independent interest in the sense that it gives rise to a new insight on well-connected H-minor-free graphs. In particular, this class of graphs can be easily decomposed into two parts so that one part has bounded treewidth and the other part is a disjoint union of bounded-genus graphs. Moreover, we can control the number of edges between the two parts. The proof method itself tells us how knowledge of a local structure can be used to gain a global structure, which gives new insight on how to decompose a graph with the help of local-structure information.

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2- and 3-factors of graphs on surfaces

Kawarabayashi

Ozeki

2010

J. Graph Theory

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It has been conjectured that any 5-connected graph embedded in a surface with sufficiently large face-width is hamiltonian. This conjecture was verified by Yu for the triangulation case, but it is still open in general. The conjecture is not true for 4-connected graphs. In this article, we shall study the existence of 2-and 3-factors in a graph embedded in a surface . A hamiltonian cycle is a special case of a 2-factor. Thus, it is quite natural to consider the existence of these factors. We give an evidence to the conjecture in a sense of the existence of a 2-factor. In fact, we only need the 4-connectivity with minimum degree at least 5. In addition, our face-width condition is not huge. Specifically, we prove the following two results. Let G be a graph embedded in a surface of Euler genus g.(1) If G is 4-connected and minimum degree of G is at least 5, and furthermore, face-width of G is at least 4g −12, then G has a 2-factor. (2) If G is 5-connected and face-width of G is at least max{44g −117, 5}, then G has a 3-factor.

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Locally planar graphs are 5-choosable

Abstract: It is proved that every graph embedded in a fixed surface with sufficiently large edge-width is 5-choosable.

Cited by 30 publications

References 16 publications

Graphs with Two Crossings Are 5-Choosable

Graphs with Two Crossings Are 5-Choosable

Additive Approximation Algorithms for List-Coloring Minor-Closed Class of Graphs

2- and 3-factors of graphs on surfaces

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