Vertex-disjoint cycles in tournaments

TL;DR

This paper extends the known results on vertex-disjoint cycles in tournaments, proving that tournaments with minimum out-degree at least 2r-1 contain at least r such cycles, generalizing previous findings.

Contribution

It generalizes prior results by establishing that tournaments with minimum out-degree at least 2r-1 contain at least r vertex-disjoint directed cycles.

Findings

Proves the existence of r vertex-disjoint cycles in tournaments with minimum out-degree ≥ 2r-1.

Extends previous results from regular to more general tournaments.

Provides a broader condition guaranteeing 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. Bessy, Sereni and Lichiardopol proved that a regular tournament $T$ of minimum degree $2r-1$ contains at least $r$ vertex-disjoint directed cycles, which shows that the above conjecture is true for tournaments. After that, Lichiardopol improved this result by showing that a $2r-1$-regular tournament contains at least $\frac{7}{6}r-\frac{7}{3}$ vertex-disjoint directed cycles. In this paper, we will extend the result to tournaments with minimum out-degree at least $2r-1$ by proving a more general result.