Abstract Sullivan stated the conjectures: (1) every oriented graph has a vertex x such that d ++(x) ≥ d −(x) and (2) every oriented graph has a vertex x such that d ++(x) + d + (x) ≥ 2d −(x). In this paper, we prove that these conjectures hold for local tournaments. In particular, for a local tournament D, there are at least two vertices satisfying (1) and either there exist two vertices satisfying (2) or there exists a vertex v satisfying d ++(v) + d + (v) ≥ 2d −(v) + 2 if D has no vertex of in-degree zero.

中文翻译：

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本地锦标赛的第二个街区

摘要 Sullivan 陈述了猜想：(1) 每个有向图都有一个顶点 x 使得 d ++(x) ≥ d −(x) 和 (2) 每个有向图都有一个顶点 x 使得 d ++(x) + d + (x) ≥ 2d -(x)。在本文中，我们证明这些猜想适用于本地锦标赛。特别地，对于本地锦标赛 D，至少有两个顶点满足 (1) 并且要么存在两个顶点满足 (2) 要么存在一个顶点 v 满足 d ++(v) + d + (v) ≥ 2d −(v) + 2 如果 D 没有入度为零的顶点。