{
  "id": "2006.13632",
  "version": "v1",
  "published": "2020-06-24T11:12:05.000Z",
  "updated": "2020-06-24T11:12:05.000Z",
  "title": "Higher matching complexes of complete graphs and complete bipartite graphs",
  "authors": [
    "Anurag Singh"
  ],
  "comment": "14 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.AT"
  ],
  "abstract": "For $r\\geq 1$, the $r$-matching complex of a graph $G$, denoted $M_r(G)$, is a simplicial complex whose faces are the subsets $H \\subseteq E(G)$ of the edge set of $G$ such that the degree of any vertex in the induced subgraph $G[H]$ is at most $r$. In this paper, we show that the complexes $M_{n-2}(K_n)$ and $M_{n-1}(K_{n,n})$ are homotopy equivalent to a wedge of spheres, where $K_n$ denotes the complete graph and $K_{m,n}$ denotes the complete bipartite graph. We also show that the complex $M_{r}(K_{m,n})$ is shellable whenever for $ m > r \\geq n$. In each case, we give a closed form formula for their homotopy types.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-24T11:12:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "55P10",
      "55U10"
    ],
    "keywords": [
      "complete bipartite graph",
      "higher matching complexes",
      "complete graph",
      "homotopy equivalent",
      "homotopy types"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}