Contributions to Seymour’s second neighborhood conjecture

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Abstract

Let [math] be a simple digraph without loops or digons. For any [math] let [math] be the set of all nodes at out-distance 1 from [math] and let [math] be the set of all nodes at out-distance 2. We show that if the underlying graph is triangle-free, there must exist some [math] such that [math] . 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.

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