$2$-colourability of the maximum ranked elements of a combinatorially sphere-like ranked poset

TL;DR

This paper generalizes a classical theorem about 2-colourability of polygonal 2-spheres with even degree vertices to higher dimensions using combinatorially sphere-like ranked posets.

Contribution

It introduces the concept of combinatorially sphere-like ranked posets and proves a 2-colourability result for their maximum elements under certain conditions.

Findings

Maximum ranked elements are 2-colourable if each element of rank (k-2) is covered by an even number of elements.

The result generalizes classical 2-sphere colourability to higher-dimensional posets.

Provides a new framework for understanding colourability in combinatorial topology.

Abstract

We obtain a higher dimensional analogue of a classical theorem which states that a polygonally cellulated $2$-sphere in $\mathbb{R}^3$, such that each vertex has even degree, is $2$-face-colourable. In order to formulate our result, we introduce the notion of combinatorially sphere-like ranked posets, which are ranked posets that generalise combinatorial spheres. We prove that, in a combinatorially sphere-like ranked poset $S$ of rank $k$, if each element of rank $(k-2)$ is covered by an even number of elements, then the maximum ranked elements of $S$ admit a proper $2$-colouring, i.e., any two adjacent maximum ranked elements have different colours.