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

TL;DR

This paper proves that certain plane graphs with degree restrictions can be facially uniquely maximum 4-colored, advancing understanding of coloring constraints in plane graphs.

Contribution

It establishes that plane graphs with vertices of degree at least four forming a star forest are facially unique-maximum 4-colorable, improving previous results for subcubic graphs.

Findings

Plane graphs with high-degree vertices forming a star forest are 4-colorable.

Disproves the general conjecture that all plane graphs are 4-colorable under facial unique-maximum coloring.

Provides new insights into coloring restrictions based on vertex degree structures.

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 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, Lidick\'y, Lu\v{z}ar, and \v{S}krekovski (2018). We conclude the paper by proposing some problems.