  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.
