Box complexes and homotopy theory of graphs

TL;DR

This paper develops a model structure on graphs that aligns with $bZ_2$-spaces, linking graph homomorphisms to $bZ_2$-homotopy equivalences via box complexes, advancing the algebraic topology approach to graph coloring.

Contribution

It introduces a new model structure on graphs that is Quillen equivalent to $bZ_2$-spaces, connecting graph homotopy theory with topological methods for coloring problems.

Findings

A model structure on graphs is established.

Weak equivalences correspond to $bZ_2$-homotopy equivalences of box complexes.

The approach provides a new perspective on the universality problem of the Hom complex.

Abstract

We introduce a model structure on the category of graphs, which is Quillen equivalent to the category of $\mathbb{Z}_2$-spaces. A weak equivalence is a graph homomorphism which induces a $\mathbb{Z}_2$-homotopy equivalence between their box complexes. The box complex is a $\mathbb{Z}_2$-space associated to a graph, considered in the context of the graph coloring problem. In the proof, we discuss the universality problem of the Hom complex.