On k-strong and k-cyclic digraphs

Thomassen (1991) proved that there is no degree of strong connectivity which guarantees a cycle through two given vertices in a digraph. In this paper we consider a large family of digraphs, including symmetric digraphs (i.e. digraphs obtained from undirected graphs by replacing each edge by a directed cycle of length two), semicomplete bipartite digraphs, locally semicomplete digraphs and all digraphs that can be obtained from acyclic digraphs and those mentioned above, by repeated substitutions of digraphs from one of these classes for vertices. We prove that for every natural number k, every k-strong digraph D from the family above is k-cyclic, i.e. for every set X of k vertices of D, there exists a cycle of D containing all the vertices of X. In particular, this implies that every k-strong quasi-transitive digraph is k-cyclic. We prove that if X is a set of vertices in a k-strong digraph D such that the maximum size of an independent set in the digraph induced by X is at most k, then D has a collection of disjoint cycles (possibly one) covering all the vertices of X. This generalizes a previous result by Jackson and Ordaz (1985). Finally, we consider k-cyclic semicomplete multipartite digraphs (SMD). We conjecture that every k-strong SMD is k-cyclic and provide some support for this conjecture.