Campus Units

Mathematics

Document Type

Article

Publication Version

Submitted Manuscript

Publication Date

6-21-2018

Journal or Book Title

arXiv

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.

Comments

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." arXiv preprint arXiv:1806.07432(2018). Posted with permission.

Copyright Owner

The Authors

Language

en

File Format

application/pdf

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