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

A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and G\"{o}ring (2016) proposed a strengthening of the Four Color Theorem conjecturing that...

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Bibliographic Details
Published in:arXiv.org 2018-06
Main Authors: Lidický, Bernard, Messerschmidt, Kacy, Riste Škrekovski
Format: Article
Language:English
Subjects:
Apexes
Color
Coloring
Four color problem
Graph theory
Graphs
Integers
Online Access:Get full text
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Summary:A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and G\"{o}ring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial unique-maximum coloring using four colors. This conjecture has been disproven for general plane graphs and it was shown that five colors suffice. In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially unique-maximum 4-colorable. This improves a previous result for subcubic plane graphs by Andova, Lidický, Lužar, and Škrekovski (2018). We conclude the paper by proposing some problems.
ISSN:2331-8422
DOI:10.48550/arxiv.1806.07432