The second out-neighborhood for local tournaments

Abstract

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.