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# Homology, Homotopy and Applications

## Volume 19 (2017)

### Number 2

### Box complexes and homotopy theory of graphs

Pages: 175 – 197

DOI: https://dx.doi.org/10.4310/HHA.2017.v19.n2.a10

#### Author

#### 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.

#### Keywords

graph, neighborhood complex, box complex, Hom complex, model category

#### 2010 Mathematics Subject Classification

Primary 55U10. Secondary 05C15.

Received 13 June 2016

Received revised 21 December 2016

Published 18 October 2017