12 problems

Let $D=(V,A)$ be a possibly infinite directed graph. An independent set is a set containing no pair joined by a directed edge. A quasi-kernel of a digraph is an independent set $K$…

A digraph is clique-acyclic if every clique has a vertex whose inneighborhood contains all other vertices of the clique. An undirected graph is kernel-solvable if every clique-acyc…

Mixed weighted 2-kernel conjecture. In every digraph $G$, and for every map $f:V(G)\rightarrow\mathbb{Z}_+$, there is a 2-kernel $K$ such that

Weighted large 2-kernel conjecture. In every digraph $G$, and for every map $f:V(G)\rightarrow\mathbb{Z}_+$, there is a 2-kernel $K$ such that

Strongly connected $k$-kernel conjecture. For all integers $k\ge 3$, every strongly connected digraph $G$ has a $k$-kernel of size at most

Large 2-kernel conjecture. In every digraph $G$, there is a 2-kernel $K$ such that at least half the vertices of $G$ belong to $K$ or have an in-neighbour in $K$.

Kernel conjecture. Every arc-coloured digraph whose every cycle is properly coloured has a kernel by properly coloured paths.

Rainbow-kernel conjecture. Every arc-colored digraph in which all cycles are rainbow has a rainbow kernel.

Bai–Feng–Zhang conjecture. Every arc-colored digraph in which all cycles are properly colored has a PCP-kernel.

$(k+1)$-kernel conjecture. Every $k$-quasi-transitive digraph has a $(k+1)$-kernel.

Minimum-cycle-color conjecture. Then $D$ has a $k$-colored kernel.

Quasi-transitive colored-kernel conjecture. $D$ has a $k$-colored kernel for every $k\geq l+1$.