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

Date
2018-06-21
Authors
Lidicky, Bernard
Lidicky, Bernard
Messerschmidt, Kacy
Škrekovski, Riste
Journal Title
Journal ISSN
Volume Title
Publisher
Source URI
Altmetrics
Authors
Research Projects
Organizational Units
Mathematics
Organizational Unit
Journal Issue
Series
Abstract

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 Göring (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.

Description
<p>This is a pre-print of the article Lidický, Bernard, Kacy Messerschmidt, and Riste Škrekovski. "Facial unique-maximum colorings of plane graphs with restriction on big vertices." <em>arXiv preprint. arXiv:1806.07432v1 </em>(2018). Posted with permission.</p>
Keywords
facial unique-maximum coloring, plane graph
Citation
Collections