Neighborhood complexes and Kronecker double coverings

Takahiro Matsushita

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arXiv · 1404.1549

Neighborhood complexes and Kronecker double coverings

Takahiro Matsushita

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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.

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Takahiro Matsushita. 2014-04-06. Neighborhood complexes and Kronecker double coverings. https://arxiv.org/abs/1404.1549

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