Contributions to Seymour's Second Neighborhood Conjecture
Abstract
Let D be a simple digraph without loops or digons. For any v in 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 provide sufficient conditions under which there must exist some v in V(D) such that |N1(v)| is less than or equal to |N2(v)|, as well as examine properties of a minimal graph which does not have such a node. We show that if one such graph exists, then there exist infinitely many strongly-connected graphs having no such vertex.
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