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New results on openly disjoint cycles in regular digraphs — arXiv:2610.41372

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.41372
License (as archived): CC0-1.0
Archived: 2026-11-02T08:00:09.634Z
SHA-256 of the archived text: e6b89710cb7a99e9030504703ceea76bcfba28172efdea34400a9a2cfb7fbf15
Stable URL of this page: https://metraipolis.com/documento/dc_6284082004b7

How to cite

New results on openly disjoint cycles in regular digraphs — arXiv:2610.41372 — arXiv (en), https://arxiv.org/abs/2610.41372 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_6284082004b7 (Metraipolis archive dc_6284082004b7, sha256 e6b89710cb7a99e9030504703ceea76bcfba28172efdea34400a9a2cfb7fbf15)

BibTeX

@misc{dc6284082004b7,
  title = {New results on openly disjoint cycles in regular digraphs — arXiv:2610.41372},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.41372},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_6284082004b7; sha256 e6b89710cb7a99e9030504703ceea76bcfba28172efdea34400a9a2cfb7fbf15},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.41372. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 \[ \left\lceil \frac{r}{6}\right\rceil \le c_r \le 5\left\lceil \frac{r}{6}\right\rceil.\]

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2610.41372 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-02T07:42:45.735Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).