Second Neighborhood via First Neighborhood in Digraphs

Abstract.

Let D be a simple digraph without loops or digons. For any $v\in V(D)$, the first out-neighborhood N ⁺(v) is the set of all vertices with out-distance 1 from v and the second neighborhood N ⁺⁺(v) of v is the set of all vertices with out-distance 2 from v. We show that every simple digraph without loops or digons contains a vertex v such that $|N^{++}(v)|\geq\gamma|N^+(v)|$, where γ = 0.657298... is the unique real root of the equation 2x ³ + x ² -1 = 0.