Unique-maximum edge-colouring of plane graphs with respect to faces

Fabrici, Igor; Jendroľ, Stanislav; Vrbjarová, Michaela

doi:10.1016/j.dam.2014.12.002

Cited by 11 publications

(11 citation statements)

References 12 publications

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“…We start by recalling an auxiliary Lemma from . Lemma (Fabrici and Göring , Lemma 6]) Let G be a plane graph with no parallel edges, let

x y \in E (G)

be an edge of G incident with the outer face, and let

c \in {black, blue}

.…”

Section: Unique Maximum 5‐coloringmentioning

confidence: 99%

“…We start by recalling an auxiliary Lemma from . Lemma (Fabrici and Göring , Lemma 6]) Let G be a plane graph with no parallel edges, let

x y \in E (G)

be an edge of G incident with the outer face, and let

c \in {black, blue}

. There is a nonproper 3‐vertex‐coloring of G with colors red, blue, and black such that (1)vertex x has color c , (2)vertex y is black, (3)each edge is incident with at most one blue vertex, (4)no vertex incident with the outer face is red, (5)each inner face is incident with at most one red vertex, and (6)each inner face that is not incident with a red vertex is incident with exactly one blue vertex. …”

Section: Unique Maximum 5‐coloringmentioning

confidence: 99%

“…A refined proof of the Four Color Theorem was given by Robertson, Seymour, Sanders, and Thomas . Our work is motivated by a conjecture of Fabrici and Göring , which, if true, would strengthen the Four Color Theorem. Conjecture (Fabrici and Göring , Conjecture 9]) Every plane graph has a proper coloring using the numbers 1, 2, 3, and 4 such that every face contains a unique vertex colored with the maximal color appearing on that face.…”

Section: Introductionmentioning

confidence: 98%

“…Note that Conjecture 1.1 holds for triangulations since any proper coloring of a triangulation has the required properties. Fabrici and Göring proved that every plane graph has a capital coloring using colors 1, …, 6. We prove a stronger result that every plane graph has a capital coloring using colors 1, …, 5.…”

Section: Introductionmentioning

confidence: 99%

“…Conjecture 1.1. (Fabrici and Göring [2], Conjecture 9]) Every plane graph has a proper coloring using the numbers 1, 2, 3, and 4 such that every face contains a unique vertex colored with the maximal color appearing on that face.…”

Section: Introductionmentioning

confidence: 99%

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Coloring of Plane Graphs with Unique Maximal Colors on Faces

Wendland

2015

Journal of Graph Theory

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warwick.ac.uk/lib-publicationsOriginal citation: Wendland, Alex. (2015) Colouring of plane graphs with unique maximal colours on faces. Journal of Graph Theory. Permanent WRAP URL:http://wrap.warwick.ac.uk/73823 Copyright and reuse:The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.Copies of full items can be used for personal research or study, educational, or not-for profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Publisher's statement:"This is the peer reviewed version of the following article: Wendland, Alex. (2015) Colouring of plane graphs with unique maximal colours on faces. Journal of Graph Theory., which has been published in final form http://dx.doi.org/10.1002/jgt.22002 . This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for SelfArchiving." A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP URL' above for details on accessing the published version and note that access may require a subscription.For more information, please contact the WRAP Team at: wrap@warwick.ac.uk Colouring of plane graphs with unique maximal colours on facesAlex Wendland * AbstractThe Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colours. Fabrici and Göring conjectured the following stronger statement to also hold: the vertices of every plane graph can be properly coloured with the numbers 1, . . . , 4 in such a way that every face contains a unique vertex coloured with the maximal colour appearing on that face. They proved that every plane graph has such a colouring with the numbers 1, . . . , 6. We prove that every plane graph has such a colouring with the numbers 1, . . . , 5 and we also prove the list variant of the statement for lists of sizes seven.

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“…We start by recalling an auxiliary Lemma from . Lemma (Fabrici and Göring , Lemma 6]) Let G be a plane graph with no parallel edges, let

x y \in E (G)

be an edge of G incident with the outer face, and let

c \in {black, blue}

.…”

Section: Unique Maximum 5‐coloringmentioning

confidence: 99%

“…We start by recalling an auxiliary Lemma from . Lemma (Fabrici and Göring , Lemma 6]) Let G be a plane graph with no parallel edges, let

x y \in E (G)

be an edge of G incident with the outer face, and let

c \in {black, blue}

Section: Unique Maximum 5‐coloringmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Coloring of Plane Graphs with Unique Maximal Colors on Faces

Wendland

2015

Journal of Graph Theory

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On the Facial Thue Choice Number of Plane Graphs Via Entropy Compression Method

Przybyło

Schreyer

Škrabuľáková

2015

Graphs and Combinatorics

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Let G be a plane graph. A vertex-colouring ϕ of G is called facial non-repetitive if for no sequence r 1 r 2 . . . r 2n , n ≥ 1, of consecutive vertex colours of any facial path it holds r i = r n+i for all i = 1, 2, . . . , n. A plane graph G is facial non-repetitively l-choosable if for every list assignment L : V → 2 N with minimum list size at least l there is a facial non-repetitive vertex-colouring ϕ with colours from the associated lists. The facial Thue choice number, π f l (G), of a plane graph G is the minimum number l such that G is facial non-repetitively l-choosable. We use the so-called entropy compression method to show that π f l (G) ≤ c∆ for some absolute constant c and G a plane graph with maximum degree ∆. Moreover, we give some better (constant) upper bounds on π f l (G) for special classes of plane graphs.

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On facial unique-maximum (edge-)coloring

Andova

Lidický

Lužar

et al. 2018

Discrete Applied Mathematics

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A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face α the maximal color appears exactly once on the vertices of α. If the coloring is required to be proper, then the upper bound for the minimal number of colors required for such a coloring is set to 5. Fabrici and Göring [5] even conjectured that 4 colors always suffice. Confirming the conjecture would hence give a considerable strengthening of the Four Color Theorem. In this paper, we prove that the conjecture holds for subcubic plane graphs, outerplane graphs and plane quadrangulations. Additionally, we consider the facial edge-coloring analogue of the aforementioned coloring and prove that every 2-connected plane graph admits such a coloring with at most 4 colors.

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Unique-maximum edge-colouring of plane graphs with respect to faces

Cited by 11 publications

References 12 publications

Coloring of Plane Graphs with Unique Maximal Colors on Faces

Coloring of Plane Graphs with Unique Maximal Colors on Faces

On the Facial Thue Choice Number of Plane Graphs Via Entropy Compression Method

On facial unique-maximum (edge-)coloring

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