The Bermond-Thomassen conjecture asserts that a finite loopless directed graph with no repeated arcs
and minimum outdegree at least 
contains 
vertex-disjoint directed cycles, for every positive
integer 
. Oppositely directed arcs are
allowed, so directed cycles of length 2 are included.
Bang-Jensen et al. (2026) proved an asymptotic version. For an absolute constant 
and all sufficiently large 
, minimum outdegree at least
suffices. This does not settle the conjectured threshold 
. Bang-Jensen et al. (2026) report AI assistance
in refining error estimates and the martingale construction.
