Directed domination in oriented hypergraphs

ErdÖs [On Schutte problem, Math. Gaz. 47 (1963)] proved that every tournament on n vertices has a directed dominating set of at most log(n+1) vertices, where log is the logarithm to base 2. He also showed that there is a tournament on n vertices with no directed domination set of cardinality less than logn−2loglogn+1. This notion of directed domination number has been generalized to arbitrary graphs by Caro and Henning in [Directed domination in oriented graphs, Discrete Appl. Math. (2012) 160:7--8.]. However, the generalization to directed r-uniform hypergraphs seems to be rare. Among several results, we prove the following upper and lower bounds on Γ→r−1(H(n,r)), the upper directed (r−1)-domination number of the complete r-uniform hypergraph on n vertices H(n,r), which is the main theorem of this paper: c(lnn)1r−1≤Γ→r−1(H(n,r))≤Clnn, where r is a positive integer and c=c(r) 0 and C=C(r) 0 are constants depending on r.