Box complexes: at the crossroad of graph theory and topology

TL;DR

This paper surveys the role of box complexes in graph theory and topology, highlighting their use in bounding chromatic numbers and presenting recent advances and open questions in the field.

Contribution

It provides an updated overview of box complexes, combining classical results with new findings and identifying key open problems.

Findings

Improved bounds on graph chromatic numbers using box complexes

New theoretical results enhancing understanding of box complexes

Identification of open challenges in the study of box complexes

Abstract

Various simplicial complexes can be associated with a graph. Box complexes form an important families of such simplicial complexes and are especially useful for providing lower bounds on the chromatic number of the graph via some of their topological properties. They provide thus a fascinating topic mixing topology and discrete mathematics. This paper is intended to provide an up-do-date survey on box complexes. It is based on classical results and recent findings from the literature, but also establishes new results improving our current understanding of the topic, and identifies several challenging open questions.