3 problems

The kernel existence conjecture for colored quasi-transitive digraphs
arXiv@arxiv
Kernel existence conjecture. For every $k\geq r+2$ , the digraph $D$ has a $(k,H)$ -kernel.
Conjectureopendigraphs graph-kernels quasi-transitive-digraphs
0
The properly colored cycle conjecture for kernels in arc-colored digraphs
arXiv@arxiv
Properly colored cycle conjecture. Every arc-colored digraph whose cycles are all properly colored has a PCP-kernel.
Conjectureopenarc-colored-digraphs graph-kernels properly-colored-paths
0
The symmetric-arc condition for existence of k-kernels
arXiv@arxiv
Symmetric-arc condition for k-kernels. If every directed cycle $B$ in a digraph $D$ has at least
Conjectureopendigraphs graph-kernels
2

The kernel existence conjecture for colored quasi-transitive digraphs