$3$ -反循环有向图是$α$ -双完全图和BE-双完全图
$3$-anti-circulant digraphs are $α$-diperfect and BE-diperfect
摘要：
设$D$是一个有向图。$V(D)$的子集$S$是稳定集，如果$S$中的每对顶点在$D$中不相邻。$D$的不相交路径$\mathcal{P}$的集合是$V(D)$的路径划分，如果$V(D)$中的每个顶点恰好在$\mathcal{P}$的路径上。我们说一个稳定集$S$和一个路径划分$\mathcal{P}$是正交的，如果$P$的每条路径恰好包含$S$的一个顶点。一个有向图$D$满足$α$ -性质，如果对于$D$的每个最大稳定集$S$，存在一个路划分$\mathcal{P}$使得$S$与$\mathcal{P}$正交.一个有向图$D$是$α$ -双完全图，如果$D$的每个导出子图都满足$α$ -性质. 1982年，Claude Berge利用禁止反有向奇圈给出了$α$ -双完全有向图的一个刻画. 2018年，Sambinelli，Silva和Lee提出了类似的猜想。有向图$D$满足Begin-End-property或BE-property，如果对于$D$的每个最大稳定集$S$，存在路径划分$\mathcal{P}$使得（i）$S$和$\mathcal{P}$正交，并且（ii）对于每个路径$P \in \mathcal{P}$，$P$的开始或结束属于$S$。有向图$D$是BE-双完全图，如果$D$的每个导出子图都满足BE-性质. Sambinelli，Silva和Lee利用禁止阻塞奇圈刻画了BE-双完全有向图。本文证明了$3$ -反循环有向图的这两个性质。我们也给出了$α$ -双完全和BE-双完全有向图的一些结构结果。
英文摘要：
Let $D$ be a digraph. A subset $S$ of $V(D)$ is a stable set if every pair of vertices in $S$ is non-adjacent in $D$. A collection of disjoint paths $\mathcal{P}$ of $D$ is a path partition of $V(D)$, if every vertex in $V(D)$ is exactly on a path of $\mathcal{P}$. We say that a stable set $S$ and a path partition $\mathcal{P}$ are orthogonal if each path of $P$ contains exactly one vertex of $S$. A digraph $D$ satisfies the $α$-property if for every maximum stable set $S$ of $D$, there exists a path partition $\mathcal{P}$ such that $S$ and $\mathcal{P}$ are orthogonal. A digraph $D$ is $α$-diperfect if every induced subdigraph of $D$ satisfies the $α$-property. In 1982, Claude Berge proposed a characterization for $α$-diperfect digraphs in terms of forbidden anti-directed odd cycles. In 2018, Sambinelli, Silva and Lee proposed a similar conjecture. A digraph $D$ satisfies the Begin-End-property or BE-property if for every maximum stable set $S$ of $D$, there exists a path partition $\mathcal{P}$ such that (i) $S$ and $\mathcal{P}$ are orthogonal and (ii) for each path $P \in \mathcal{P}$, either the start or the end of $P$ belongs to $S$. A digraph $D$ is BE-diperfect if every induced subdigraph of $D$ satisfies the BE-property. Sambinelli, Silva and Lee proposed a characterization for BE-diperfect digraphs in terms of forbidden blocking odd cycles. In this paper, we verified both conjectures for $3$-anti-circulant digraphs. We also present some structural results for $α$-diperfect and BE-diperfect digraphs.
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