5 problems

Seymour's second neighbourhood conjecture. Every orientation $G$ contains at least one vertex $v$ such that

Sullivan's conjecture. Every oriented graph contains at least one vertex $v$ such that

Converse-invariance conjecture. If

Approximate Seymour conjecture. For every $\epsilon>0$, every oriented graph $G$ has at least one vertex $v$ satisfying

Let $k$ be a positive integer. For a graph $G$ and a vertex $x\in V(G)$, let $N_G(x)$ be the neighbourhood of $x$, and let $N_G^2(x)$ be the set of vertices other than $x$ sharing…