## Abstract

In 2003, A. Björner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SG_{n,k} is homotopy equivalent to a k-sphere. Further, for n = 2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of SG_{n,k} contains as a deformation retract the boundary complex of a simplicial polytope. Our purpose is to give a positive answer to this question in the case k = 2. We also find in this case that, after partially subdividing the neighborhood complex, the resulting complex deformation retracts onto a subcomplex arising as a polyhedral boundary sphere that is invariant under the action induced by the automorphism group of SG_{n,2}.

Original language | English |
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Pages (from-to) | 413-427 |

Number of pages | 15 |

Journal | Proceedings of the American Mathematical Society |

Volume | 142 |

Issue number | 2 |

DOIs | |

State | Published - 2014 |

## Keywords

- Discrete morse theory
- Neighborhood complex
- Polytope
- Stable kneser graph

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics