Deformation retracts of neighborhood complexes of stable Kneser graphs

Abstract

In 2003, A. Bjorner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SGn,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 SGn,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 SGn,2.