A characterisation of 3-colourable 3-dimensional triangulations

TL;DR

This paper extends Heawood's theorem to 3-dimensional triangulations, establishing that such a triangulation is 3-edge-colorable if and only if all edges have even degree, thus characterizing 3-colorability in 3D.

Contribution

It provides a necessary and sufficient condition for 3-colorability of 3D triangulations, expanding classical planar results to three dimensions.

Findings

Triangulations of 3-space are 3-edge-colorable iff all edges have even degree.

Generalizes Heawood's theorem from planar to 3D triangulations.

Establishes a clear criterion linking edge degrees to colorability in 3D.

Abstract

We extend Heawood's theorem on the colourability of plane triangulations to triangulations of 3-space. We prove that a triangulation of 3-space can be edge coloured with three colours if and only if all edges have even degree.