Some Results on Berge’s Conjecture and Begin–End Conjecture

Let D be a digraph. A subset S of V(D) is stable if every pair vertices in non-adjacent D. collection disjoint paths $$\mathcal {P}$$ path partition V(D), vertex on . We say that set and are orthogonal contains exactly one S. digraph satisfies the $$\alpha $$ -property for maximum D, there to -diperfect induced subdigraph -property. In 1982, Berge proposed characterization digraphs terms forbidden anti-directed odd cycles. 2018, Sambinelli, Silva Lee similar conjecture. Begin–End-property or BE-property such (i) (ii) each $$P\in \mathcal , either initial final P lies BE-diperfect BE-property. blocking this paper, we show some structural results digraphs. particular, minimal counterexample both conjectures, it follows (D) < \vert V(D)\vert /2$$ Moreover, prove conjectures arc-locally (out) in-semicomplete