Colouring of plane graphs with unique maximal colours on faces

TL;DR

This paper proves that every plane graph can be coloured with five colours to ensure each face has a unique maximal colour, improving previous bounds and extending to list colourings with seven colours.

Contribution

It establishes the existence of a 5-colouring with unique face-maximal colours for all plane graphs, improving the previous 6-colour bound and providing a list variant with 7 colours.

Findings

Every plane graph admits a 5-colouring with unique face-maximal colours.

The list variant holds for lists of size 7.

Improves the previous bound of 6 colours for this property.

Abstract

The Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colors. Fabrici and G\"oring conjectured the following stronger statement to also hold: the vertices of every plane graph can be properly coloured with the numbers 1,...,4 in such a way that every face contains a unique vertex coloured with the maximal color appearing on that face. They proved that every plane graph has such a colouring with the numbers 1,...,6. We prove that every plane graph has such a colouring with the numbers 1,...,5 and we also prove the list variant of the statement for lists of sizes seven.