On facial unique-maximum (edge-)coloring

TL;DR

This paper investigates facial unique-maximum colorings of plane graphs, proving the conjecture that four colors suffice for certain classes and extending results to facial edge-colorings with up to four colors.

Contribution

It confirms the conjecture for subcubic, outerplane, and quadrangulated plane graphs, and extends the concept to facial edge-colorings with four colors.

Findings

Four colors suffice for facial unique-maximum colorings of subcubic plane graphs.

The conjecture holds for outerplane graphs and plane quadrangulations.

Facial edge-colorings of 2-connected plane graphs can be achieved with at most four colors.

Abstract

A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. If the coloring is required to be proper, then the upper bound for the minimal number of colors required for such a coloring is set to $5$. Fabrici and G\"oring [Fabrici and Goring 2016] even conjectured that $4$ colors always suffice. Confirming the conjecture would hence give a considerable strengthening of the Four Color Theorem. In this paper, we prove that the conjecture holds for subcubic plane graphs, outerplane graphs and plane quadrangulations. Additionally, we consider the facial edge-coloring analogue of the aforementioned coloring and prove that every $2$-connected plane graph admits such a coloring with at most $4$ colors.