论文标题
超级条件迫使挖掘中的短周期
Outdegree conditions forcing short cycles in digraphs
论文作者
论文摘要
给定一个正整数$ m \ ge 3 $，让$ ch（m）$是以下属性的最小正常数： \ emph {$ n \ ge 3 $顶点上的每一个简单的导向图，所有其超级均为$ ch（m）\ cdot n $最多包含$ m $长度的定向周期。} Caccetta和Häggkvist猜想$ ch（m）= 1/m $，如果是的话，将是最好的。在本文中，我们证明了以下结果： \ emph {对于每个整数$ m \ ge 3 $，令$α（m）$是公式的$（0,1）$中唯一的真实根，} \ begin {equination*}（1-x）（1-x）^{m-2} = \ frac {3x} {3x} {2-x} {2-x}。 \ end {equation*}然后$ ch（m）\leα（m）$。 这概括了沉的结果。 然后，我们使用Chudnovsky，Seymour和Sullivan启动的最小反馈弧集方法稍微改善了上述不等式。这导致了汉堡，哈克塞尔和科斯托卡的发现的扩展（在情况下为$ m = 3 $），以及梁和Xu（在情况下$ m = 4 $）。
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 $m=4$).
