Neighborhood complexes and Kronecker double coverings

TL;DR

This paper explores the relationship between neighborhood complexes and Kronecker double coverings of graphs, showing how isomorphic coverings imply isomorphic complexes and constructing examples with specific chromatic properties.

Contribution

It establishes a connection between Kronecker double coverings and neighborhood complexes, including new constructions of graphs with isomorphic coverings but different chromatic numbers.

Findings

Kronecker double coverings being isomorphic implies neighborhood complexes are isomorphic

Constructed graphs with identical neighborhood complexes but different chromatic numbers

Found non-isomorphic graphs with isomorphic Kronecker double coverings

Abstract

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.