{
  "id": "2412.01121",
  "version": "v1",
  "published": "2024-12-02T04:47:44.000Z",
  "updated": "2024-12-02T04:47:44.000Z",
  "title": "Transversal Structures in Graph Systems: A Survey",
  "authors": [
    "Wanting Sun",
    "Guanghui Wang",
    "Lan Wei"
  ],
  "comment": "24 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a system $\\mathcal{G} =\\{G_1,G_2,\\dots,G_m\\}$ of graphs/digraphs/hypergraphs on the common vertex set $V$ of size $n$, an $m$-edge graph/digraph/hypergraph $H$ on $V$ is transversal in $\\mathcal{G}$ if there exists a bijection $\\phi :E(H)\\rightarrow [m]$ such that $e \\in E(G_{\\phi(e)})$ for all $e\\in E(H)$. In this survey, we consider extremal problems for transversal structures in graph systems. More precisely, we summarize some sufficient conditions that ensure the existence of transversal structures in graph/digraph/hypergraph systems, which generalize several classical theorems in extremal graph theory to transversal version. We also include a number of conjectures and open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-02T04:47:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "transversal structures",
      "common vertex set",
      "extremal graph theory",
      "open problems",
      "transversal version"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}