arXiv 2008.091712020-08-20math.CO

有向图中强迫短圈的出度条件

Outdegree conditions forcing short cycles in digraphs

Dan Ismailescu, Joonsoo Lee, Andrew Yang

摘要：

给定一个正整数$m\ge 3$，让$ch(m)$是最小的正常数，并具有以下性质：\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和Häggkvist证明了$ch(m)=1/m$，如果为真，将是最好的可能。本文证明了以下结果：\emph{ For every integer $m\ge 3$, let $α(m)$ be the unique real root in $(0,1)$ of the equation} \begin{equation*} (1-x)^{m-2}=\frac{3x}{2-x}. \end{equation*}然后是$ch(m)\le α(m)$ .这推广了Shen证明了$ch(3)\le 3-\sqrt{7}<0.35425$，Liang和Xu证明了$ch(4)< 0.28866$和$ch(5)<0.24817$的结果.然后，我们稍微改善上述不等式使用的最小反馈弧集的方法发起的Chudnovsky，西摩，沙利文。这导致了Hamburger、Haxell和Kostochka（在$m=3$的情况下）以及Liang和Xu（在$m=4$的情况下）的发现的扩展。

英文摘要：

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 Häggkvist 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 $α(m)$ be the unique real root in $(0,1)$ of the equation} \begin{equation*} (1-x)^{m-2}=\frac{3x}{2-x}. \end{equation*} Then $ch(m)\le α(m)$. 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 $m=4$).

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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