Coloring of Plane Graphs with Unique Maximal Colors on Faces

Abstract

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.

On facial unique-maximum (edge-)coloring

“…Conjecture 3 (Wendland [9]). If each vertex of a plane graph is assigned a list of 5 integers, then there exists a FUM-coloring assigning each vertex a color from its list.…”

A counterexample to a conjecture on facial unique-maximal colorings

“…as the maximum of χ fum (G) over all graphs G embedded into Σ. Our construction and the result of Wendland [6] implies that χ fum (S 0 ) = 5, where S 0 is the sphere. Our result motivates to study this invariant for graphs on other surfaces.…”

“…Andova, Lidický, Lužar, and Škrekovski [1] showed that 4 colors suffice for outerplanar graphs and for subcubic plane graphs. Wendland [6] also considered the list coloring version of the problem, where he was able to prove the upper bound 7 and conjectured that lists of size 5 are sufficient. Edge version of the problem was considered by Fabrici, Jendrol', and Vrbjarová [5].…”

“…When stating the conjecture, Fabrici and Göring [4] proved that χ fum (G) ≤ 6 for every plane graph G. Promptly, this coloring was considered by others. Wendland [6] decreased the upper bound to 5 for all plane graphs. Andova, Lidický, Lužar, and Škrekovski [1] showed that 4 colors suffice for outerplanar graphs and for subcubic plane graphs.…”

Facial unique-maximum colorings of plane graphs with restriction on big vertices

“…This conjecture was disproven in the general case by the authors [5]. Fabrici and Göring [4] proved that for any plane graph G, χ fum (G) ≤ 6, while Wendland [8] improved the upper bound to 5. Andova, Lidický, Lužar, and Škrekovski [1] proved that if G is a subcubic or outerplane graph, χ fum (G) ≤ 4.…”