Hom complexes and homotopy theory in the category of graphs

Abstract

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 Lovász 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.

28 pages, 13 figures, final version, to be published in European J. Comb

References in corpus (3)

• Simple homotopy types of Hom-complexes, neighborhood complexes, Lovász complexes, and atom crosscut complexes
• Collapsing along monotone poset maps
• Homotopy groups of Hom complexes of graphs

Cited by in corpus (1)

• The Discrete Fundamental Group of the Order Complex of $B_n$