Seymour's Second Neighborhood Conjecture for orientations of (pseudo)random graphs

Abstract

Seymour's Second Neighborhood Conjecture (SNC) states that every oriented graph contains a vertex whose second neighborhood is as large as its first neighborhood. We investigate the SNC for orientations of both binomial and pseudo random graphs, verifying the SNC asymptotically almost surely (a.a.s.) (i) for all orientations of $G(n,p)$ if $\limsup_{n\to\infty} p < 1/4$; and (ii) for a uniformly-random orientation of each weakly $(p,A\sqrt{np})$-bijumbled graph of order $n$ and density $p$, where $p=\Omega(n^{-1/2})$ and $1-p = \Omega(n^{-1/6})$ and $A>0$ is a universal constant independent of both $n$ and $p$. We also show that a.a.s. the SNC holds for almost every orientation of $G(n,p)$. More specifically, we prove that a.a.s. (iii) for all $\varepsilon > 0$ and $p=p(n)$ with $\limsup_{n\to\infty} p \le 2/3-\varepsilon$, every orientation of $G(n,p)$ with minimum outdegree $\Omega_\varepsilon(\sqrt{n})$ satisfies the SNC; and (iv) for all $p=p(n)$, a random orientation of $G(n,p)$ satisfies the SNC.