AI 中文总结
该研究证明了最小出度为$\delta$、阶数$n=2\delta+2$且缺失边无规定结构的定向图满足Seymour第二邻域猜想，将反例阶数下界从16提至17，条件下提至19。
AI 中文摘要
Seymour第二邻域猜想断言：每个有限定向图都存在一个顶点，其精确第二出邻域的数量至少与出邻域的数量相等。已证明的情形包括：锦标赛图，由Fisher于1996年证明；最小出度至多为6的定向图，由Kaneko和Locke于2001年证明；Sadhukhan、Sandeep和Sen于2026年的最新预印本研究了最小出度为7的情况。对于稠密不完全图，Fidler和Yuster于2007年证明了当缺失边构成匹配、星或团时猜想成立，Ghazal于2012年将这一方向扩展到广义星，Dara、Francis、Jacob和Narayanan于2022年证明了当缺失边可划分为一个匹配和一个星时猜想成立。我们对每个阶数为$n=2\delta+2$（其中$\delta$为最小出度）且缺失边无规定结构的定向图，给出了该猜想的简短计数证明。结合Fisher的锦标赛定理，这意味着所有满足$n\le2\delta+2$的定向图都满足该猜想。结合已知的最小出度结果，这将我们所知的反例阶数的最佳下界从16提高到17，并且在Sadhukhan、Sandeep和Sen（2026）预印本成立的条件下，从18提高到19。
英文摘要
Seymour's second neighborhood conjecture asserts that every finite oriented graph has a vertex with at least as many exact second outneighbors as outneighbors. Established cases include tournaments, proved by Fisher (1996), and oriented graphs of minimum outdegree at most six, proved by Kaneko and Locke (2001); a recent preprint of Sadhukhan, Sandeep, and Sen (2026) treats minimum outdegree seven. For dense incomplete graphs, Fidler and Yuster (2007) proved the conjecture when the missing edges form a matching, a star, or a clique, Ghazal (2012) extended this direction to generalized stars, and Dara, Francis, Jacob, and Narayanan (2022) proved it when the missing edges can be partitioned into a matching and a star. We give a short counting proof of the conjecture for every oriented graph of order $n=2δ+2$, where $δ$ is the minimum outdegree, with no prescribed structure on the missing edges. Together with Fisher's tournament theorem, this implies the conjecture for every oriented graph satisfying $n\le2δ+2$. Combined with the known minimum-outdegree results, this raises the best lower bound known to us on the order of a counterexample from $16$ to $17$ and, conditional on the preprint of Sadhukhan, Sandeep, and Sen (2026), from $18$ to $19$.
Comments5 pages
