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PreprintCC BY 4.0arXiv

An asymptotic bound for the Bermond--Thomassen Conjecture

  • Jørgen Bang-Jensen
  • Guanghui Wang
  • Yun Wang
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Abstract

The Bermond--Thomassen Conjecture asserts that every digraph with minimum out-degree at least 2k−1 contains k disjoint directed cycles; it remains open in general. We asymptotically resolve it: there exist an absolute constant K and a function g(k)=o(k), independent of n, such that every n-vertex finite simple loopless digraph with minimum out-degree at least 2k+g(k) contains k disjoint directed cycles for all integers k≥ K and n≥1. In fact we obtain the explicit error term g(k)=O(k3/4√(log k)).

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Cite

BibTeXRIS
BibTeX
@misc{bangjensen2026asymptotic,
  title = {{An asymptotic bound for the Bermond--Thomassen Conjecture}},
  author = {Jørgen Bang-Jensen and Guanghui Wang and Yun Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11517},
  eprint = {2610.11517},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11517},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An asymptotic bound for the Bermond--Thomassen Conjecture
AU  - Jørgen Bang-Jensen
AU  - Guanghui Wang
AU  - Yun Wang
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11517
UR  - https://arxiv.org/abs/2610.11517
AB  - The Bermond--Thomassen Conjecture asserts that every digraph with minimum out-degree at least 2k−1 contains k disjoint directed cycles; it remains open in general. We asymptotically resolve it: there exist an absolute constant K and a function g(k)=o(k), independent of n, such that every n-vertex finite simple loopless digraph with minimum out-degree at least 2k+g(k) contains k disjoint directed cycles for all integers k≥ K and n≥1. In fact we obtain the explicit error term g(k)=O(k^3/4√(log k)).
ER  -