Colouring simplicial complexes via the Lechuga-Murillo's model

TL;DR

This paper extends Lechuga and Murillo's graph colouring model to simplicial complexes, providing algebraic criteria for colourability based on Sullivan algebras, thus broadening the algebraic-topological approach to combinatorial problems.

Contribution

It generalizes the algebraic graph colouring model to simplicial complexes, introducing new notions of colourings and associated Sullivan algebras.

Findings

A Sullivan algebra criterion for simplicial complex colourability

Extension of graph colouring algebraic models to higher-dimensional complexes

New notions of colourings for simplicial complexes

Abstract

L. Lechuga and A. Murillo showed that a non-oriented, simple, connected, finite graph $G$ is $k$-colourable if and only if a certain pure Sullivan algebra associated to $G$ and $k$ is not elliptic. In this paper, we extend this result to simplicial complexes by means of several notions of colourings of these objects.