Outdegree conditions forcing short cycles in digraphs

TL;DR

This paper establishes upper bounds on the outdegree conditions that guarantee short directed cycles in graphs, generalizing previous results and improving bounds through feedback arc set techniques.

Contribution

It proves a new upper bound for the outdegree condition ensuring short cycles in digraphs, extending and refining prior bounds for all m ≥ 3.

Findings

Bound ch(m) ≤ α(m) for all m ≥ 3, where α(m) is a specific root.

Generalizes previous bounds for m=3,4,5.

Improves bounds using feedback arc set approach.

Abstract

Given a positive integer $m\ge 3$, let $ch(m)$ be the smallest positive constant with the following property: \emph{ Every simple directed graph on $n\ge 3$ vertices all whose outdegrees are at least $ch(m)\cdot n$ contains a directed cycle of length at most $m$.} Caccetta and H\"{a}ggkvist conjectured that $ch(m)=1/m$, which if true, would be the best possible. In this paper, we prove the following result: \emph{ For every integer $m\ge 3$, let $\alpha(m)$ be the unique real root in $(0,1)$ of the equation} \begin{equation*} (1-x)^{m-2}=\frac{3x}{2-x}. \end{equation*} Then $ch(m)\le \alpha(m)$. This generalizes results of Shen who proved that $ch(3)\le 3-\sqrt{7}<0.35425$, and Liang and Xu who showed that $ch(4)< 0.28866$ and $ch(5)<0.24817$. We then slightly improve the above inequality by using the minimum feedback arc set approach initiated by Chudnovsky, Seymour, and Sullivan. This results in extensions of the findings of Hamburger, Haxell and Kostochka (in the case $m=3$), and Liang and Xu (in the case $m=4$).