Contributions to Seymour's second neighborhood conjecture

Abstract

Let $D$ be a simple digraph without loops or digons. For any $v\in V\left(D\right)$ let $N_1\left(v\right)$ be the set of all nodes at out-distance 1 from $v$ and let $N_2\left(v\right)$ be the set of all nodes at out-distance 2. We show that if the underlying graph is triangle-free, there must exist some $v\in V\left(D\right)$ such that $\left|N_1\left(v\right)\right|\le \left|N_2\left(v\right)\right|$. We provide several properties a “minimal” graph which does not contain such a node must have. Moreover, we show that if one such graph exists, then there exist infinitely many.