A minimum degree condition forcing a digraph to be k-linked

Given a digraph D, let δ 0 ( D ) : = min { δ + ( D ) , δ − ( D ) } be the minimum semi-degree of D. In [D. Kühn and D. Osthus, Linkedness and ordered cycles in digraphs, submitted] we showed that every sufficiently large digraph D with δ 0 ( D ) ≥ n / 2 + ℓ − 1 is ℓ-linked. The bound on the minimum semi-degree is best possible and confirms a conjecture of Manoussakis [Y. Manoussakis, k-linked and k-cyclic digraphs, J. Combinatorial Theory B 48 (1990) 216-226]. We [D. Kühn and D. Osthus, Linkedness and ordered cycles in digraphs, submitted] also determined the smallest minimum semi-degree which ensures that a sufficiently large digraph D is k-ordered, i.e. that for every sequence s 1 , … , s k of distinct vertices of D there is a directed cycle which encounters s 1 , … , s k in this order.