The second out-neighbourhood for local tournaments

Abstract

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