{
  "abstract": "The basis number $bn(G)$ of a graph $G$ is the minimum edge-congestion of a basis of its cycle space. We prove that every finite $n$-vertex multigraph satisfies $bn(G)=O(\\log n)$, resolving, for simple graphs, a question of Bazargani, Biedl, Bose, Maheshwari and Miraftab, subsequently stated as a conjecture by Miraftab, Morin and Yuditsky. The argument also yields the cycle-rank refinement $bn(G)=O(\\log β(G))$, where $β(G)$ is the dimension of the cycle space, and a reduction of Lehner and Miraftab, based on a theorem of Richter and Shank, then gives $bn(G)=O(\\log g)$ for graphs of Euler genus $g$. For regular matroids we prove the ground-set bound $bn(M)=O(\\log m)$, where $m=|E(M)|$, and logarithmic bounds in both the rank $r(M)$ and the cycle-space dimension $d$. All these orders are best possible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02080",
  "author_total": 1,
  "authors": [
    {
      "name": "Kolja Knauer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2pqtbm1tk5k61.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02080",
  "fulltext_url": null,
  "id": "2pqtbm1tk5k61",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02080",
  "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|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7207822612",
  "pdf_url": "https://arxiv.org/pdf/2609.02080",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2pqtbm1tk5k61.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.02080",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02080"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Logarithmic basis number of graphs and regular matroids",
  "title_text": "Logarithmic basis number of graphs and regular matroids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/2pqtbm1tk5k61",
  "venue": "arXiv",
  "versions": []
}