Bermond–Thomassen conjecture for k greater than or equal to 4

Prove that every digraph D with minimum out-degree at least 2k−1 contains k vertex-disjoint directed cycles for every integer k≥4.

Background

The Bermond–Thomassen conjecture asserts that a minimum out-degree of 2k−1 forces k vertex-disjoint directed cycles in a digraph. The conjecture is known for k=2 and k=3, while the paper explicitly identifies its validity for every k≥4 as unresolved. The main theorem of the paper proves only a relaxation in which each cycle may be made directed by reversing one arc, so it does not settle the original conjecture.

References

So far, the precise form of the conjecture has only been proved for~$k=2$ by Thomassen in the same~$1983$ article, and more recently for~$k=3$ by Lichiardopol, P{o}r and Sereni. It remains wide open for each~$k\ge 4$, and only for restricted classes, such as tournaments, has the conjecture been proved in full generality.

— A relaxation of the Bermond-Thomassen conjecture  (2608.12948 - Bessy et al., 13 Aug 2026) in Section Introduction

In the same paper, Thomassen also conjectured that the precise bound $b(k)=2k-1$ holds for every $k$; the same conjecture in fact appeared already earlier in the survey article on cycles in digraphs by Bermond and Thomassen in 1981, and is nowadays famous as the (still open) Bermond-Thomassen conjecture.

— Proof of Lichiardopol's conjecture on disjoint directed cycles of distinct lengths  (2608.20012 - Albrechtsen et al., 20 Aug 2026) in Introduction