{
  "abstract": "Let $Φ(G)$ denote the number of perfect matchings of a graph $G$. Goedgebeur et al. posted two questions about the perfect matchings of planar 3-connected graphs: does there exist an infinite family of matchable planar 3-connected graphs of girth 5 with a constant number of perfect matchings? Whether there exist infinitely many planar 3-connected graphs with minimum degree 5 which have a constant number of perfect matchings. We present answers to those questions. First, we construct, for every even $n\\ge196$, a simple planar 3-connected graph of order $n$ and girth five with exactly $c$ perfect matchings, where $c$ is a fixed positive integer. Second, for a simple planar 3-connected graph $G$ with a perfect matching and minimum degree at least five, we show that $Φ(G)\\ge2^{\\lceil(|V(G)|+20)/22\\rceil}$. This implies that, for each positive integer $t$, only finitely many such graphs, up to isomorphism, have at most $t$ perfect matchings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16648",
  "author_total": 1,
  "authors": [
    {
      "name": "Xinyu Dai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b7akgwcnegjmq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16648",
  "fulltext_url": null,
  "id": "b7akgwcnegjmq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16648",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221803337",
  "pdf_url": "https://arxiv.org/pdf/2610.16648",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b7akgwcnegjmq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16648",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16648"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Perfect matchings in planar 3-connected graphs: girth five and minimum degree five",
  "title_text": "Perfect matchings in planar 3-connected graphs: girth five and minimum degree five",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/b7akgwcnegjmq",
  "venue": "arXiv",
  "versions": []
}