Homotopy types of box complexes

In [14] Matoušek and Ziegler compared various topological lower bounds for the chromatic number. They proved that Lovász’s original bound [9] can be restated as X(G) ≥ ind(B(G)) + 2. Sarkaria’s bound [15] can be formulated as X(G) ≥ ind(B0(G)) + 1. It is known that these lower bounds are close to each other, namely the difference between them is at most 1. In this paper we study these lower bounds, and the homotopy types of box complexes. The most interesting result is that up to ℤ2-homotopy the box complex B(G) can be any ℤ2-space. This together with topological constructions allows us to construct graphs showing that the mentioned two bounds are different. Some of the results were announced in [14].