Hom complexes and homotopy theory in the category of graphs

We investigate a notion of $\times$-homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph $\times$-homotopy is characterized by the topological properties of the $\Hom$ complex, a functorial way to assign a poset (and hence topological space) to a pair of graphs; $\Hom$ complexes were introduced by Lov\'{a}sz and further studied by Babson and Kozlov to give topological bounds on chromatic number. Along the way, we also establish some structural properties of $\Hom$ complexes involving products and exponentials of graphs, as well as a symmetry result which can be used to reprove a theorem of Kozlov involving foldings of graphs. Graph $\times$-homotopy naturally leads to a notion of homotopy equivalence which we show has several equivalent characterizations. We apply the notions of $\times$-homotopy equivalence to the class of dismantlable graphs to get a list of conditions that again characterize these. We end with a discussion of graph homotopies arising from other internal homs, including the construction of `$A$-theory' associated to the cartesian product in the category of reflexive graphs.