Note on the chromatic number of Minkowski planes: the regular polygon case

TL;DR

This paper investigates the chromatic number of Minkowski planes with polygonal unit circles, establishing an upper bound of 6 colors for regular polygons with up to 22 vertices, extending previous results.

Contribution

It introduces a simple lattice-sublattice coloring scheme that proves an upper bound of 6 colors for these Minkowski planes, a novel result for polygons with more than 8 vertices.

Findings

Chromatic number of Minkowski planes with regular polygonal unit circles is at most 6.

The coloring scheme applies to polygons with up to 22 vertices.

New bounds established for polygons with more than 8 vertices.

Abstract

The famous Hadwiger-Nelson problem asks for the minimum number of colors needed to color the points of the Euclidean plane so that no two points unit distance apart are assigned the same color. In this note we consider a variant of the problem in Minkowski metric planes, where the unit circle is a regular polygon of even and at most 22 vertices. We present a simple lattice-sublattice coloring scheme that uses 6 colors, proving that the chromatic number of the Minkowski planes above are at most 6. This result is new for regular polygons having more than 8 vertices.