Lichiardopol's conjecture on disjoint cycles in tournaments

TL;DR

This paper proves Lichiardopol's conjecture that tournaments with sufficiently high minimum out-degree contain a specified number of disjoint cycles of a given length, completing the proof for all cases.

Contribution

The paper confirms Lichiardopol's conjecture for all cycle lengths $q eq 3,4$, extending previous results and completing the proof.

Findings

Confirmed the conjecture for $q eq 3,4$

Established minimum out-degree conditions for disjoint cycles

Completed the proof of Lichiardopol's conjecture

Abstract

In 2010, N. Lichiardopol conjectured for $q \geq 3$ and $k \geq 1$ that any tournament with minimum out-degree at least $(q-1)k-1$ contains $k$ disjoint cycles of length $q$. We prove this conjecture for $q \geq 5$. Since it is already known to hold for $q\le4$, this completes the proof of the conjecture.