A counterexample to a conjecture on facial unique-maximal colorings

TL;DR

This paper disproves a conjecture that four colors suffice for facial unique-maximum colorings of plane graphs by providing an infinite family of counterexamples, establishing that five colors are necessary.

Contribution

It introduces an infinite family of counterexamples to the conjecture, showing that the facial unique-maximum chromatic number can be five, not four.

Findings

Counterexamples with five colors are necessary for facial unique-maximum colorings.

The conjecture that four colors suffice is false.

Facial unique-maximum chromatic number of the sphere is five.

Abstract

A facial unique-maximum coloring of a plane graph is a proper vertex coloring by natural numbers where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. Fabrici and G\"oring proved that six colors are enough for any plane graph and conjectured that four colors suffice. This conjecture is a strengthening of the Four Color theorem. Wendland later decreased the upper bound from six to five. In this note, we disprove the conjecture by giving an infinite family of counterexamples. Thus we conclude that facial unique-maximum chromatic number of the sphere is five.