{
  "id": "2310.06354",
  "version": "v1",
  "published": "2023-10-10T06:43:57.000Z",
  "updated": "2023-10-10T06:43:57.000Z",
  "title": "Transversals in a collections of trees",
  "authors": [
    "Ethan Y. H. Li",
    "Luyi Li",
    "Ping Li"
  ],
  "comment": "16pages,2figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathcal{S}$ be a fixed family of graphs on vertex set $V$ and $\\mathcal{G}$ be a collection of elements in $\\mathcal{S}$. We investigated the transversal problem of finding the maximum value of $|\\mathcal{G}|$ when $\\mathcal{G}$ contains no rainbow elements in $\\mathcal{S}$. Specifically, we determine the exact values when $\\mathcal{S}$ is a family of stars or a family of trees of the same order $n$ with $n$ dividing $|V|$. Further, all the extremal cases for $\\mathcal{G}$ are characterized.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-10T06:43:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C05",
      "05D15"
    ],
    "keywords": [
      "collection",
      "exact values",
      "vertex set",
      "rainbow elements",
      "maximum value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}