exists g:N→N forall k∈N forall D bigl(delta^+(D)≥ g(k) Longrightarrow exists C_1,…,C_k vertex-disjoint directed cycles in D such that |C_i|≠|C_j| for all i≠ j bigr), where delta^+(D) denotes the minimum out-degree of D.

Lichiardopol’s conjecture

The conjecture asks whether, for every $k \ge 1$ , some minimum out-degree bound forces $k$ vertex-disjoint directed cycles with pairwise distinct lengths. Earlier literature records that even the case $k=3$ remained open.

Known results

Strongly $s(k)$ -connected digraphs satisfy the conclusion.
With sufficiently large minimum in-degree as well as minimum out-degree, the conclusion holds for $k=3$ .
For directed tree-width at most $d$ , the bound $\delta^+(D)>(d+2)(k-1)$ suffices.
The conjecture was partially verified for tournaments, regular digraphs, and digraphs of small order.

August 2026 claimed proof

An unrefereed arXiv manuscript claims the full conjecture, using butterfly minors, directed tangles, and directed flat-wall theorems. The scan found no independent exposition, verification, or published corroboration.

Current status (as of August 2026): The conjecture has a new unrefereed claimed proof, but its correctness is not yet independently established; absent that claim, only partial results are settled.

Lichiardopol’s conjecture

Primary source

Additional references

Progress summary

Solutions 0