About colorings, stability and paths in directed graphs

AbstractIn [2] Berge asked: In a digraph, does there exist an optimal coloring and a path meeting each color class exactly once? We give a negative answer to this question when the chromatic number is ⩾5. He also asked: Does every directed graph G have a maximum stable set (with ∣S∣=α(G), the stability number of the digraph) and a partition of vertex-set into paths μ1, μ2,…, μα(G) such that ∣μi∩S∣=1 for all i? We give here a negative answer to this question