2 problems

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The kernel existence conjecture for colored quasi-transitive digraphs
Kernel existence conjecture. For every $k\geq r+2$ , the digraph $D$ has a $(k,H)$ -kernel.
Conjectureopendigraphs graph-kernels quasi-transitive-digraphsarXiv posted 2 days ago
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The $(k+1)$ -kernel conjecture for $k$ -quasi-transitive digraphs
$(k+1)$ -kernel conjecture. Every $k$ -quasi-transitive digraph has a $(k+1)$ -kernel.
Conjectureopendigraphs kernels quasi-transitive-digraphsarXiv posted 2 days ago

The kernel existence conjecture for colored quasi-transitive digraphs