  In [MZ04] Matousek and Ziegler compared various topological lower bounds for
the chromatic number. They proved that Lovasz's original bound [L78] can be
restated as $\chr G \geq \ind (\B(G)) +2$. Sarkaria's bound [S90] can be
formulated as $\chr G \geq \ind (\B_0(G)) +1$. 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. Some of the results was announced in [MZ04].
