Homotopy types of box complexes

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].Comment: 11 page

Generalization of neighborhood complexes

We introduce the notion of r-neighborhood complex for a positive integer r, which is a natural generalization of Lovasz neighborhood complex. The topologies of these complexes give some obstructions of the existence of graph maps. We applied these complexes to prove the nonexistence of graph maps about Kneser graphs. We prove that the fundamental groups of r-neighborhood complexes are closely related to the (2r)-fundamental groups defined in the author's previous paper.Comment: 8 page

Neighborhood complexes and Kronecker double coverings

The neighborhood complex $N(G)$ is a simplicial complex assigned to a graph $G$ whose connectivity gives a lower bound for the chromatic number of $G$. We show that if the Kronecker double coverings of graphs are isomorphic, then their neighborhood complexes are isomorphic. As an application, for integers $m$ and $n$ greater than 2, we construct connected graphs $G$ and $H$ such that $N(G) \cong N(H)$ but $\chi(G) = m$ and $\chi(H) = n$. We also construct a graph $KG_{n,k}'$ such that $KG_{n,k}'$ and the Kneser graph $KG_{n,k}$ are not isomorphic but their Kronecker double coverings are isomorphic.Comment: 10 pages. Some results concerning box complexes are deleted. to appear in Osaka J. Mat

Transitivity is not a (big) restriction on homotopy types

For every simplicial complex K there exists a vertex-transitive simplicial complex homotopy equivalent to a wedge of copies of K with some copies of the circle. It follows that every simplicial complex can occur as a homotopy wedge summand in some vertex-transitive complex. One can even demand that the vertex-transitive complex is the clique complex of a Cayley graph or that it is facet-transitive

On topological relaxations of chromatic conjectures

There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number