{
  "abstract": "Galluccio--Loebl and Tesler showed that the perfect-matching polynomial of a graph embedded in an orientable surface of genus $g$ can be written as a linear combination of at most $4^g$ Pfaffians. We show that, in general, exponentially many Pfaffians are necessary. More precisely, for every $g\\ge1$, there exists a graph of orientable genus at most $g$ whose perfect-matching polynomial requires at least $(8/3)^g$ Pfaffians in any such linear representation. In particular, for every even integer $n\\ge6$, there is a graph on $n$ vertices with Pfaffian number at least $(8/3)^{\\lfloor n/6\\rfloor}$. Moreover, the lower bound is witnessed even by cubic bipartite matching-covered graphs. We prove this by showing that expressing the permanent of an $n\\times n$ matrix of distinct variables as a linear combination of determinants obtained by changing signs of its entries requires exponentially many determinants. As a consequence, we improve a recent linear lower bound on the Pfaffian number due to Junchaya, Miranda, and Lucchesi to an exponential lower bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.21077",
  "author_total": 2,
  "authors": [
    {
      "name": "Priyanshu Pant"
    },
    {
      "name": "Ranveer Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6yy876zr09ptm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.21077",
  "fulltext_url": null,
  "id": "6yy876zr09ptm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.21077",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7162038818",
  "pdf_url": "https://arxiv.org/pdf/2605.21077",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6yy876zr09ptm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2605.21077",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.21077"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exponential Lower Bounds for the Pfaffian Number of Graphs",
  "title_text": "Exponential Lower Bounds for the Pfaffian Number of Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/6yy876zr09ptm",
  "venue": "arXiv",
  "versions": []
}