7 problems

Let $D$ be a non-empty digraph. For a vertex $u\in V(D)$, let $d^+(u)$ denote its out-degree, let $N^+[u]$ denote its closed out-neighborhood, and call a vertex a source if it has…

Worst-case output-size conjecture. For every $\varepsilon>0$, there exists a sourceless digraph $D$ such that, for every vertex $v$ of $D$, every quasi-kernel returned by the algor…

Out-degree-three expansion conjecture. There exists a vertex $v\in V(D)$ such that

Small Quasi-Kernel Conjecture. The digraph $D$ contains a quasi-kernel of order at most $\frac{n}{2}$.

Kostochka–Luo–Shan conjecture. Every digraph $D$ has a quasi-kernel of size at most

Erdős–Hajnal–Soukup partition conjecture. There is a partition

Erdős–Soukup's conjecture. Every digraph $G=(V,E)$ has a partition $(V_0,V_1)$ such that $G[V_0]$ has a quasi-kernel and $G[V_1]$ has a quasi-sink.