7 problems

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

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 $\frac{n}{2}$.

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

Gutin–Koh–Tay–Yeo's conjecture. Every sink-free digraph has two disjoint quasi-kernels.

Let $D$ be an $n$-vertex digraph, and let $S$ be the set of sources of $D$. For a set $X\subseteq V(D)$, let $N_D^+(X)$ denote the set of outneighbors of vertices in $X$ outside…