Optimal minimum-out-degree function for distinct-length cycle packings

Determine the smallest function g:N→N such that every digraph of minimum out-degree at least g(k) contains k vertex-disjoint directed cycles of pairwise distinct lengths.

Background

The main theorem proves that some function g with this property exists, thereby resolving Lichiardopol’s existence conjecture. However, the proof does not determine the optimal threshold function.

The authors give a quadratic lower bound, (k²+3k−2)/2, based on the minimum number of vertices required by k vertex-disjoint directed cycles of distinct lengths and the corresponding complete-digraph construction. They leave the problem of finding the smallest valid function open.