Department of
Applied
Mathematics

A new bound for a particular case of the Caccetta-Häggkvist conjecture

Nicolas Lichiardopol

Lycée Adam de Craponne, Salon, Francie

Abstract

In a recent paper, Hamburger, Haxell and Kostochka have proved that if $\alpha \geq 0.35312$, then any digraph $D$ of order $n$ with minimum out-degree at least $\alpha n$, contains a directed cycle of length at most three. They have also proved that if $\beta \geq 0.34564$, then any digraph $D$ of order $n$ with both minimum out-degree and minimum in-degree at least $\beta n$, contains a directed cycle of length at most three. In my talk, by using the first result, I slightly improve the second bound. Namely, I prove that if $\beta \geq \frac{1}{2.9016}\approx 0.34464$, then any digraph $D$ of order $n$ with both minimum out-degree and minimum in-degree at least $\beta n$, contains a directed cycle of length at most three.