{
  "abstract": "The pfaffian number of a graph $G$, denoted by $\\mathrm{pf}(G)$, is the minimum integer $k$ such that the generating function of perfect matchings of $G$ can be written as a linear combination of $k$ pfaffians of signed adjacency matrices. The Galluccio-Loebl-Tesler Theorem states that $\\mathrm{pf}(G) \\leq 4^g$, where $g$ is the orientable genus of $G$. We generalize this theorem by relating the pfaffian number of a graph to another graph parameter: the symbolic pfaffian number, denoted by $\\mathrm{spf}(G)$. Namely, we prove that $\\mathrm{pf}(G) \\leq 2^{\\mathrm{spf}(G)}$. Our proof is based on linear algebra arguments and does not depend on topological properties of graphs. As a consequence of our generalization, we conclude that $\\mathrm{pf}(K_{2n}) = 2^{\\mathcal{O}(n \\log n)}$, an improvement on the previous upper bound of $4^{\\mathcal{O}(n^2)}$ based on the genus. We also consider lower bounds for $\\mathrm{pf}(G)$ related to $\\mathrm{spf}(G)$. In particular, we show that the graphs with $\\mathrm{pf}(G) = 4$ are the graphs with $\\mathrm{spf}(G) = 2$. On the other hand, we show that our main result is strictly more general than a previous generalization by Norine.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00468",
  "author_total": 3,
  "authors": [
    {
      "name": "Enrique Junchaya"
    },
    {
      "name": "Cláudio L. Lucchesi"
    },
    {
      "name": "Alberto A. A. Miranda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2c658qzvz5vdz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00468",
  "fulltext_url": null,
  "id": "2c658qzvz5vdz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00468",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00468",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2c658qzvz5vdz.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00468",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00468"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The symbolic pfaffian method I: A generalization of the Gallucio-Loebl-Tesler Theorem",
  "title_text": "The symbolic pfaffian method I: A generalization of the Gallucio-Loebl-Tesler Theorem",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/2c658qzvz5vdz",
  "venue": "arXiv",
  "versions": []
}