k-Ordered Hamilton cycles in digraphs

Given a digraph D, let δ0(D):=min{δ+(D),δ−(D)} be the minimum semi-degree of D. D is k-ordered Hamiltonian if for every sequence s1,…,sk of distinct vertices of D there is a directed Hamilton cycle which encounters s1,…,sk in this order. Our main result is that every digraph D of sufficiently large order n with δ0(D)⩾⌈(n+k)/2⌉−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, Sárközy and Selkow [H. Kierstead, G. Sárközy, S. Selkow, On k-ordered Hamiltonian graphs, J. Graph Theory 32 (1999) 17–25].