Contributions to Seymour’s second neighborhood conjecture

Let D be a simple digraph without loops or digons. For anyv2 V.D/ let N1.v/ be the set of all nodes at out-distance 1 from v and let N2.v/ be the set of all nodes at out-distance 2. We show that if the underlying graph is triangle-free, there must exist somev2 V.D/ such thatjN1.v/jj N2.v/j. 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.