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A conjecture by Lichiardopol states that for every $k \ge 1$ there exists aninteger $g(k)$ such that every digraph of minimum out-degree at least $g(k)$contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths.Motivated by Lichiardopol's conjecture, we study the existence ofvertex-disjoint directed cycles satisfying length constraints in digraphs oflarge connectivity or large minimum degree. Our main result is that for every$k \in \mathbb{N}$, there exists $s(k) \in \mathbb{N}$ such that every strongly$s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles ofpairwise distinct lengths. In contrast, for every $k \in \mathbb{N}$ weconstruct a strongly $k$-connected digraph containing no two vertex- orarc-disjoint directed cycles of the same length. It is an open problem whether$g(3)$ exists. Here we prove the existence of an integer $K$ such that everydigraph of minimum out- and in-degree at least $K$ contains $3$ vertex-disjointdirected cycles of pairwise distinct lengths.

Disjoint cycles with length constraints in digraphs of large connectivity or minimum degree