{
  "id": "2403.15298",
  "version": "v1",
  "published": "2024-03-22T15:45:09.000Z",
  "updated": "2024-03-22T15:45:09.000Z",
  "title": "On the matching complexes of categorical product of path graphs",
  "authors": [
    "Raju Kumar Gupta",
    "Sourav Sarkar",
    "Sagar S. Sawant",
    "Samir Shukla"
  ],
  "comment": "36 pages, 33 figures, comments are welcome",
  "categories": [
    "math.CO"
  ],
  "abstract": "The matching complex $\\mathsf{M}(G)$ of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. These complexes appears in various places and found applications in many areas of mathematics including; discrete geometry, representation theory, combinatorics, etc. In this article, we consider the matching complexes of categorical product $P_n \\times P_m$ of path graphs $P_n$ and $P_m$. For $m = 1$, $P_n \\times P_m$ is a discrete graph and therefore its matching complex is the void complex. For $m = 2$, $\\mathsf{M}(P_n \\times P_m)$ has been proved to be homotopy equivalent to a wedge of spheres by Kozlov. We show that for $n \\geq 2$ and $3 \\leq m \\leq 5$, the matching complex of $P_n \\times P_m$ is homotopy equivalent to a wedge of spheres. For $m =3$, we give a closed form formula for the number and dimension of spheres appearing in the wedge. Further, for $m \\in \\{4, 5\\}$, we give minimum and maximum dimension of spheres appearing in the wedge in the homotopy type of $\\mathsf{M}(P_n \\times P_m)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-22T15:45:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P10",
      "05E45",
      "55U10"
    ],
    "keywords": [
      "matching complex",
      "path graphs",
      "categorical product",
      "homotopy equivalent",
      "simplicial complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}