{
  "abstract": "For integers $n>k\\ge1$ with $nk$ even, let $m(n,k)$ be the maximum size of a graph of order $n$ with a unique $k$-factor. In 1964, Hetyei and in 1984, Hendry [\\emph{J. Combin. Theory Ser. B} \\textbf{37} (1984), 53--63] proved $m(n,1)=n^2/4$ for even $n$ and $m(n,2)=\\lfloor n(n+1)/4\\rfloor$, respectively. In 2000, Johann [\\emph{J. Graph Theory} \\textbf{35} (2000), 227--243] showed that $m(n,k)=nk/2+\\binom{n-k}{2}$ for $k>n/2$ and $kn$ even and $m(n,k)=n^2/4+(k-1)n/4$ for $n=2kq$, where $q$ is a positive integer. Later, Volkmann [\\emph{Combinatorica} \\textbf{24} (2004), 531--540] proved $m(n,k)=k^2+\\binom{n-k}{2}$ for $n/3\\le k<n/2$ and $kn$ even, and he determined $m(n,3)$. In this paper, we determine $m(n,k)$ for all admissible pairs, proving the size conjecture posed by Volkmann. We also prove that every extremal graph with $2\\le k\\le n-2$ has a universal vertex whose neighbors in the unique factor are precisely its degree-$k$ vertices. In particular, this result confirms Volkmann's structural conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00950",
  "author_total": 1,
  "authors": [
    {
      "name": "Zixuan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4p8r8pdmdjnyp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00950",
  "fulltext_url": null,
  "id": "4p8r8pdmdjnyp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00950",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00950",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4p8r8pdmdjnyp.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00950",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00950"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The size and structure of graphs with a unique $k$-factor",
  "title_text": "The size and structure of graphs with a unique k-factor",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/4p8r8pdmdjnyp",
  "venue": "arXiv",
  "versions": []
}