Disjoint high-out-degree subdigraphs

Determine whether, for every s∈N, there exists a number F(s) such that every digraph of minimum out-degree at least F(s) contains two vertex-disjoint subdigraphs, each of minimum out-degree at least s.

Background

The paper explains that one tempting strategy for proving the distinct-length cycle result would be to find several vertex-disjoint subdigraphs with sufficiently large minimum out-degree and then select cycles of different lengths from them.

This strategy depends on an independently posed problem attributed to Alon and Stiebitz. The authors emphasize that it is an old and difficult open problem, reportedly reduced in recent work to the case s=2, and that this approach remains beyond the reach of the methods used in the paper.

References

However, it turns out to be an old, amazingly difficult and well-known open problem, posed independently by Alon and Stiebitz, to determine whether there exists, for every $s\in \mathbb{N}$, a number~$F(s)$ such that every digraph of minimum out-degree at least~$F(s)$ contains two vertex-disjoint subdigraphs of minimum out-degree at least~$s$.

— Proof of Lichiardopol's conjecture on disjoint directed cycles of distinct lengths  (2608.20012 - Albrechtsen et al., 20 Aug 2026) in Paragraph “Some remarks on the proof of Theorem 2 and related problems”