Colouring the 1-skeleton of $d$-dimensional triangulations

TL;DR

This paper characterizes which triangulations of the $d$-sphere have a $(d+2)$-colourable 1-skeleton, revealing a specific divisibility condition related to subdivisions and cell incidences.

Contribution

It provides a structural characterization of $d$-sphere triangulations with $(d+2)$-colourable 1-skeletons, extending previous work on $(d+1)$-colourability.

Findings

Triangulations with a subdivision satisfying divisibility conditions have $(d+2)$-colourable 1-skeletons.

The characterization involves the incidence count of $(d-1)$-cells around $(d-2)$-cells.

This advances understanding of chromatic properties of high-dimensional triangulations.

Abstract

While every plane triangulation is colourable with three or four colours, Heawood showed that a plane triangulation is 3-colourable if and only if every vertex has even degree. In $d \geq 3$ dimensions, however, every $k \geq d+1$ may occur as the chromatic number of some triangulation of ${\mathbb S}^d$. As a first step, Joswig structurally characterised which triangulations of ${\mathbb S}^d$ have a $(d+1)$-colourable 1-skeleton. In the 20 years since Joswig's result, no characterisations have been found for any $k>d+1$. In this paper, we structurally characterise which triangulations of ${\mathbb S}^d$ have a $(d+2)$-colourable 1-skeleton: they are precisely the triangulations that have a subdivision such that for every $(d-2)$-cell, the number of incident $(d-1)$-cells is divisible by three.