Some generalizations of Mengerʹs theorem concerning arc-connected digraphs

It is proved that, if D is a k-arc-connected digraph, then (i) for any k distinct arcs fi = xiyi, i = 1, … ,k, D − fi, … ,fk contains k arc-disjoint spanning arborescences T1, … , Tk such that Ti is rooted at yi for each i, and (ii) for any k triples (x1, f1, y1), … , (xk,fk,yk), where x1 , … , xk, y1, … , yk ∈ V(D) (not necessarily distinct) and f1, … , fk are distinct arcs with fi ∈ E+ (xi), i = 1, … , k(resp. fi ∈ E−(yi), i = 1, … , k), there exist in D k arc-distinct xi − yi paths Pi such that fi ∈ E(Pi), i = 1, … , k.