关于西摩和沙利文的第二邻域猜想
On Seymour's and Sullivan's Second Neighbourhood Conjectures
AI总结：
研究西摩和沙利文关于有向图顶点邻域关系的第二邻域猜想，给出满足沙利文猜想的传递三角形数量充分条件，证明其对平面图等成立，还表明对一些定向分裂图族两个猜想也成立。
摘要：
对于有向图的一个顶点\(x\)，\(d^+(x)\)（分别地，\(d^-(x)\)）是距离\(x\)为\(1\)的出（入，分别地）顶点的数量，\(d^{++}(x)\)是距离\(x\)为\(2\)的顶点的数量。1995年，西摩猜想对于任何定向图\(D\)存在一个顶点\(x\)使得\(d^+(x)\leq d^{++}(x)\)。2006年，沙利文猜想在\(D\)中存在一个顶点\(x\)使得\(d^-(x)\leq d^{++}(x)\)。我们给出了一个关于传递三角形数量的充分条件，使得一个定向图满足沙利文猜想。特别地，这意味着沙利文猜想对于平面图和无三角形图的所有定向都成立。如果有向图\(D\)的顶点可以被划分为顶点集\(X\)和\(Y\)，使得\(X\)是一个独立集且\(Y\)诱导出一个竞赛图，那么\(D\)是一个定向分裂图。我们还表明这两个猜想对于一些定向分裂图族成立，特别是当\(Y\)诱导出一个正则或几乎正则竞赛图时。
英文摘要：
For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any oriented graph $D$ there exists a vertex $x$ such that $d^+(x)\leq d^{++}(x)$. In 2006, Sullivan conjectured that there exists a vertex $x$ in $D$ such that $d^-(x)\leq d^{++}(x)$. We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and of triangle-free graphs. An oriented graph $D$ is an oriented split graph if the vertices of $D$ can be partitioned into vertex sets $X$ and $Y$ such that $X$ is an independent set and $Y$ induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when $Y$ induces a regular or an almost regular tournament.
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