Generalization of neighborhood complexes

TL;DR

This paper generalizes Lovasz neighborhood complexes to r-neighborhood complexes, exploring their topological properties and applications in graph map obstructions, especially for Kneser graphs.

Contribution

It introduces r-neighborhood complexes, analyzes their topological features, and links their fundamental groups to (2r)-fundamental groups, providing new tools for graph theory.

Findings

r-neighborhood complexes generalize Lovasz complexes

Topologies of these complexes obstruct certain graph maps

Fundamental groups relate to (2r)-fundamental groups

Abstract

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.