{
  "id": "2407.12289",
  "version": "v1",
  "published": "2024-07-17T03:21:48.000Z",
  "updated": "2024-07-17T03:21:48.000Z",
  "title": "On intersecting families of subgraphs of perfect matchings",
  "authors": [
    "Melissa M. Fuentes",
    "Vikram Kamat"
  ],
  "comment": "10 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The seminal Erd\\H{o}s--Ko--Rado (EKR) theorem states that if $\\mathcal{F}$ is a family of $k$-subsets of an $n$-element set $X$ for $k\\leq n/2$ such that every pair of subsets in $\\mathcal{F}$ has a nonempty intersection, then $\\mathcal{F}$ can be no bigger than the trivially intersecting family obtained by including all $k$-subsets of $X$ that contain a fixed element $x\\in X$. This family is called the star centered at $x$. In this paper, we formulate and prove an EKR theorem for intersecting families of subgraphs of the perfect matching graph, the graph consisting of $n$ disjoint edges. This can be considered a generalization not only of the aforementioned EKR theorem but also of a signed variant of it, first stated by Meyer (1974), and proved separately by Deza--Frankl (1983) and Bollob\\'as--Leader (1997). The proof of our main theorem relies on a novel extension of Katona's beautiful cycle method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-17T03:21:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "05C35"
    ],
    "keywords": [
      "intersecting family",
      "perfect matching",
      "main theorem relies",
      "katonas beautiful cycle method",
      "novel extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}