{
  "abstract": "Let $M$ be a matroid and let $\\Tutte_M(x,y)$ be its Tutte polynomial. The multiplicative Merino--Welsh inequality \\[ \\Tutte_M(2,0)\\Tutte_M(0,2)\\geq \\Tutte_M(1,1)^2 \\] was proved to be false for matroids by Beke, Csáji, Csikvári, and Pituk (\\emph{Adv. Math.} \\textbf{446} (2024), 109674). Let $U_{n,r}$ denote the rank-$r$ uniform matroid on $n$ elements, where $0<r<n$, and let $U_{n,r}^{(q)}$ denote the matroid obtained by replacing each element of $U_{n,r}$ with $q$ parallel copies. They constructed an infinite family of counterexamples of the form $U_{n,r}^{(2)}$, including $U_{33,22}^{(2)}$. In this paper, we prove that every $U_{n,r}^{(q)}$ with $q\\geq3$ satisfies the inequality. For $q=2$, we prove that the minimum number of elements in such a counterexample is $62$, attained by $U_{31,21}^{(2)}$. For loopless and bridgeless graphs, Noble and Royle (\\emph{Eur. J. Comb.} \\textbf{38} (2014), 24--35) showed that every edge of a minimum-size counterexample to the multiplicative Merino--Welsh conjecture lies in a series or parallel class of size two or three. For matroids, we further prove that every element of a minimum-size loopless and coloopless counterexample lies in exactly one series or parallel class of size two.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27706",
  "author_total": 3,
  "authors": [
    {
      "name": "Jungang Chen"
    },
    {
      "name": "Xian'an Jin"
    },
    {
      "name": "Tianlong Ma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/18jce55z7m7q1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27706",
  "fulltext_url": null,
  "id": "18jce55z7m7q1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27706",
  "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",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223131671",
  "pdf_url": "https://arxiv.org/pdf/2610.27706",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/18jce55z7m7q1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27706",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27706"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On minimum-size matroid counterexamples to the multiplicative Merino--Welsh inequality",
  "title_text": "On minimum-size matroid counterexamples to the multiplicative Merino--Welsh inequality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/18jce55z7m7q1",
  "venue": "arXiv",
  "versions": []
}