{
  "abstract": "Given a digraph \\(D\\), let \\(c(D)\\) be the largest size of a collection of openly disjoint directed cycles through a common vertex, and let \\(c_r\\) be the minimum of \\(c(D)\\) over all \\(r\\)-regular digraphs \\(D\\). In 2005, Seymour asked whether every \\(r\\)-regular digraph satisfies \\(c(D)=r\\). Mader answered Seymour's problem negatively for every \\(r\\ge 8\\), but conjectured that nevertheless \\(\\lim_{r\\to\\infty}c_r=\\infty\\). Recently, Steiner confirmed this conjecture by proving \\[ \\left\\lceil \\frac{3r}{22}\\right\\rceil \\le c_r \\le 7\\left\\lceil \\frac{r}{8}\\right\\rceil. \\] In this paper, we first answer Seymour's problem negatively for $r\\in \\{6,7\\}$, and then improve Steiner's bound to \\[ \\left\\lceil \\frac{r}{6}\\right\\rceil \\le c_r \\le 5\\left\\lceil \\frac{r}{6}\\right\\rceil.\\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41372",
  "author_total": 3,
  "authors": [
    {
      "name": "Bin Chen"
    },
    {
      "name": "Wenshen Wu"
    },
    {
      "name": "Meishaung Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/74a7s17af2mn5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41372",
  "fulltext_url": null,
  "id": "74a7s17af2mn5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41372",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41372",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/74a7s17af2mn5.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41372",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41372"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "New results on openly disjoint cycles in regular digraphs",
  "title_text": "New results on openly disjoint cycles in regular digraphs",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/74a7s17af2mn5",
  "venue": "arXiv",
  "versions": []
}