A Note on Solid Coloring of Pure Simplicial Complexes

TL;DR

This paper generalizes a known planar coloring result to higher dimensions, showing that pure simplicial complexes in R^d can be colored with d+1 colors to prevent adjacent simplices from sharing the same color.

Contribution

It extends the classical map-coloring theorem to higher-dimensional simplicial complexes, providing a simple coloring method for pure complexes in R^d.

Findings

In R^2, planar maps with triangular faces are 3-colorable.

In R^3, collections of tetrahedra can be solid 4-colored.

The coloring ensures no two simplices sharing a (d-1)-face have the same color.

Abstract

We establish a simple generalization of a known result in the plane. The simplices in any pure simplicial complex in R^d may be colored with d+1 colors so that no two simplices that share a (d-1)-facet have the same color. In R^2 this says that any planar map all of whose faces are triangles may be 3-colored, and in R^3 it says that tetrahedra in a collection may be "solid 4-colored" so that no two glued face-to-face receive the same color.