{
  "abstract": "Let \\(b(M)\\), \\(d(M)\\), and \\(r(M)\\) denote the number of bases, the number of circuits, and the rank of a matroid \\(M\\), respectively. We prove that every nonempty simple binary matroid with no coloops satisfies \\[ 2b(M)\\ge(r(M)+1)d(M), \\] with equality if and only if \\(M\\) is isomorphic to the Fano matroid. This confirms a conjecture recorded by Oxley in 1983. For the same class, we prove that deleting any element leaves at least as many bases as there are circuits in the original matroid: \\(b(M\\backslash e)\\ge d(M)\\) for every \\(e\\in E(M)\\). We determine all equality cases and deduce a sharp linear lower bound for basis growth under successive series extensions. We also give a circuit-size exclusion condition guaranteeing distinct bases containing \\(C-e\\) for circuit-element pairs \\((C,e)\\) with prescribed circuit sizes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11829",
  "author_total": 1,
  "authors": [
    {
      "name": "Houshan Fu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2v2jfxsrty8yk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11829",
  "fulltext_url": null,
  "id": "2v2jfxsrty8yk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11829",
  "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",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221151063",
  "pdf_url": "https://arxiv.org/pdf/2610.11829",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2v2jfxsrty8yk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.11829",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11829"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A rank bound for bases and circuits in binary matroids",
  "title_text": "A rank bound for bases and circuits in binary matroids",
  "type": "preprint",
  "updated_at": "2026-11-03T12:21:09Z",
  "url": "https://research.draconitic.com/paper/2v2jfxsrty8yk",
  "venue": "arXiv",
  "versions": []
}