关于第二邻域猜想的一些结果的推广
Extending some results on the second neighborhood conjecture
摘要：
有向图中的顶点被称为具有大的第二邻域，如果它具有至少与外邻域一样多的第二外邻域。第二邻域猜想，首先由Seymour提出，断言在每个有向图（没有循环或对角的有向图）中存在具有大的第二邻域的顶点。我们证明了定向图的缺失的边缘可以划分成（可能是空的）匹配和（可能是空的）星满足这个猜想。这推广了Fidler和Yuster的一个结果。我们的结果的一个含义是，每一个定向图没有一个汇，其缺失的边缘形成一个（可能是空的）匹配至少有两个顶点与大的第二个邻域。这是一个加强定理的哈维特和Pillasse，谁表明，同样适用于锦标赛没有水槽。此外，我们还证明了对于顶点集可划分为一个独立集和一个2-退化图的定向图，该猜想是成立的。
英文摘要：
A vertex in a directed graph is said to have a large second neighborhood if it has at least as many second out-neighbors as out-neighbors. The Second Neighborhood Conjecture, first stated by Seymour, asserts that there is a vertex having a large second neighborhood in every oriented graph (a directed graph without loops or digons). We prove that oriented graphs whose missing edges can be partitioned into a (possibly empty) matching and a (possibly empty) star satisfy this conjecture. This generalizes a result of Fidler and Yuster. An implication of our result is that every oriented graph without a sink and whose missing edges form a (possibly empty) matching has at least two vertices with large second neighborhoods. This is a strengthening of a theorem of Havet and Thomasse, who showed that the same holds for tournaments without a sink. Moreover, we also show that the conjecture is true for oriented graphs whose vertex set can be partitioned into an independent set and a 2-degenerate graph.
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