k-Ordered Hamilton cycles in digraphs

Given a digraph D, let @d^0(D):=min{@d^+(D),@d^-(D)} be the minimum semi-degree of D. D is k-ordered Hamiltonian if for every sequence s"1,...,s"k of distinct vertices of D there is a directed Hamilton cycle which encounters s"1,...,s"k in this order. Our main result is that every digraph D of sufficiently large order n with @d^0(D)>[email protected]?(n+k)/[email protected]?-1 is k-ordered Hamiltonian. The bound on the minimum semi-degree is best possible. An undirected version of this result was proved earlier by Kierstead, Sarkozy and Selkow [H. Kierstead, G. Sarkozy, S. Selkow, On k-ordered Hamiltonian graphs, J. Graph Theory 32 (1999) 17-25].