AI 中文总结
研究西摩第二邻域猜想，提出更强版本，即存在从\(N^+_D(x)\)到\(N^{++}_D(x)\)的完美匹配，证明该更强版本对最小出度至多\(5\)的定向图和\(5 -\)反传递定向图成立，进而推出定向平面图满足此更强版本。
AI 中文摘要
西摩的一个长期猜想，即西摩第二邻域猜想，指出每个定向图\(D\)都包含一个顶点\(x\)，使得\(|N^{++}_D(x)|\geq |N^{+}_D(x)|\)。该猜想在一些特殊定向图类中得到验证，但对一般定向图仍未解决。我们提出一个更强版本的猜想，即每个定向图\(D\)都包含一个顶点\(x\)，使得存在从\(N^+_D(x)\)到\(N^{++}_D(x)\)的完美匹配。我们证明这个更强版本对每个最小出度至多为\(5\)的定向图以及每个\(5 -\)反传递定向图都成立。这意味着每个定向平面图都满足更强版本。
英文摘要
A longstanding conjecture of Seymour, called Seymour's second neighborhood conjecture, states that every oriented graph $D$ contains a vertex $x$ with $|N^{++}_D(x)|\geq |N^{+}_D(x)|$. The conjecture was verified in a few special classes of oriented graphs, and it remains open for general oriented graphs. We study a stronger property, asking for a vertex $x$ such that there exists a complete matching from $N^+_D(x)$ to $N^{++}_D(x)$. We prove that this stronger version holds for every oriented graph with minimum out-degree at most $5$, and also for every $5$-anti-transitive oriented graph. This implies that every oriented planar graph satisfies the stronger version.
