Seymour's second neighbourhood conjecture: random graphs and reductions

Abstract: 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 $p\in[0,1/2)$, 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 $p\in(1/2,1)$, 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.