Deformation Retracts of Neighborhood Complexes of Stable Kneser Graphs

Details Deformation Retracts of Neighborhood Complexes of Stable Kneser Graphs Deformation Retracts of Neighborhood Complexes of Stable Kneser Graphs

2011-02-09

arxiv.org/abs/1102.1984v1

Mathematics

Combinatorics

15 pages, 1 figure

Scientific paper

In 2003, A. Bjorner 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}.

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