Seymour's second neighbourhood conjecture: random graphs and reductions

A longstanding conjecture of Seymour states that in every oriented graph there is a vertex whose second outneighbourhood is at least as large as its outneighbourhood. In this short note we show that, for any fixed p∈[0,1/2), a.a.s. every orientation of G(n,p) satisfies Seymour's conjecture (as well as a related conjecture of Sullivan). This improves on a recent result of Botler, Moura and Naia. Moreover, we show that p=1/2 is a natural barrier for this problem, in the following sense: for any fixed p∈(1/2,1), Seymour's conjecture is actually equivalent to saying that, with probability bounded away from 0, every orientation of G(n,p) satisfies Seymour's conjecture. This provides a first reduction of the problem. For a second reduction, we consider minimum degrees and show that, if Seymour's conjecture is false, then there must exist arbitrarily large strongly-connected counterexamples with bounded minimum outdegree. Contrasting this, we show that vertex-minimal counterexamples must have large minimum outdegree.