{
  "id": "1802.05848",
  "version": "v1",
  "published": "2018-02-16T07:02:16.000Z",
  "updated": "2018-02-16T07:02:16.000Z",
  "title": "Homotopy type of Neighborhood Complexes of Kneser graphs, $KG_{2,k}$",
  "authors": [
    "Nandini Nilakantan",
    "Anurag Singh"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Schrijver identified a family of vertex critical subgraphs of the Kneser graphs called the stable Kneser graphs $SG_{n,k}$. Bj\\\"{o}rner and de Longueville 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 $SG_{2,k}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-16T07:02:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "57M15"
    ],
    "keywords": [
      "neighborhood complex",
      "homotopy type",
      "stable kneser graph",
      "homotopy equivalent",
      "vertex critical subgraphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}