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.