Vertex-disjoint directed cycles of prescribed length in tournaments with given minimum out-degree

TL;DR

This paper proves a conjecture about the existence of multiple vertex-disjoint directed cycles of prescribed length in tournaments with a given minimum out-degree, extending previous results and confirming a broader conjecture.

Contribution

It affirms Lichiardopol's conjecture on vertex-disjoint q-cycles in tournaments with specified out-degree conditions, generalizing earlier results.

Findings

Confirmed Lichiardopol's conjecture for all q ≥ 3.

Extended Bermond-Thomassen conjecture to tournaments.

Established minimum out-degree conditions ensure multiple disjoint cycles.

Abstract

The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. In 2014, Bang-Jensen, Bessy and Thomass\' e proved the conjecture for tournaments. In 2010, Lichiardopol conjectured that a tournament $T$ with minimum out-degree at least $(q-1)r-1$ contains at least $r$ vertex-disjoint $q$-cycles, where integer $q\geq3$ and $r\geq1$. In this paper, we address Lichiardopol's conjecture affirmatively. In particular, the case $q=3$ implies Bermond-Thomassen conjecture for tournaments.