Abstract
Schrijver (Nieuw Archief voor Wiskunde, 26(3) (1978) 454–461) identified a family of vertex critical subgraphs of the Kneser graphs called the stable Kneser graphs \(SG_{n,k}\). Björner and de Longueville (Combinatorica 23(1) (2003) 23–34) proved that the neighborhood complex of the stable Kneser graph \(SG_{n,k}\) is homotopy equivalent to a k-sphere. In this article, we prove that the homotopy type of the neighborhood complex of the Kneser graph \(KG_{2,k}\) is a wedge of \((k+4)(k+1)+1\) spheres of dimension k. We construct a maximal subgraph \(S_{2,k}\) of \(KG_{2,k}\), whose neighborhood complex is homotopy equivalent to the neighborhood complex of \(SG_{2,k}\). Further, we prove that the neighborhood complex of \(S_{2,k}\) deformation retracts onto the neighborhood complex of \(SG_{2,k}\).
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The authors would like to thank the referee for having provided several relevant comments.
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Nilakantan, N., Singh, A. Homotopy type of neighborhood complexes of Kneser graphs, \(\varvec{KG_{2,k}}\). Proc Math Sci 128, 53 (2018). https://doi.org/10.1007/s12044-018-0429-9
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DOI: https://doi.org/10.1007/s12044-018-0429-9