Stationary map coloring

TL;DR

This paper proves that a planar Voronoi map derived from a Poisson process can be properly colored with 6 colors in a deterministic, isometry-equivariant way, and explores related geometric properties and open questions.

Contribution

It establishes the existence of a 6-coloring for the Voronoi map as a deterministic, isometry-equivariant function of the Poisson process, and analyzes the Delaunay triangulation core.

Findings

Existence of a 6-coloring for the Voronoi map.

The 6-core of the Delaunay triangulation is empty.

Discussion of generalizations and open questions.

Abstract

We consider a planar Poisson process and its associated Voronoi map. We show that there is a proper coloring with 6 colors of the map which is a deterministic isometry-equivariant function of the Poisson process. As part of the proof we show that the 6-core of the corresponding Delaunay triangulation is empty. Generalizations, extensions and some open questions are discussed.