  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.
