Lichiardopol’s Conjecture on Disjoint Cycles in Tournaments

Abstract

In 2010, Lichiardopol conjectured for $q \geqslant 3$ and $k \geqslant 1$ that any tournament with minimum out-degree at least $(q-1)k-1$ contains $k$ disjoint cycles of length $q$. Previously the conjecture was known to hold for $q\leqslant 4$. We prove that it holds for $q \geqslant 5$, thereby completing the proof of the conjecture.