Covering a Strong Digraph by α−1 Disjoint Paths: A Proof of Las Vergnasʹ Conjecture

The Gallai–Milgram theorem states that every directed graph D is spanned by α(D) disjoint directed paths, where α(D) is the size of a largest stable set of D. When α(D)>1 and D is strongly connected, it has been conjectured by Las Vergnas that D is spanned by an arborescence with α(D)−1 leaves. The case α=2 follows from a result of C. C. Chen and P. Manalastas (1983, Discrete Math.44, 243–250). We give a proof of this conjecture in the general case.