k-Ary spanning trees contained in tournaments

A rooted tree is called a k-ary tree, if all non-leaf vertices have exactly k children, except possibly one non-leaf vertex has at most k−1 children. Denote by h(k) the minimum integer such that every tournament of order at least h(k) contains a k-ary spanning tree. It is well-known that every tournament contains a Hamiltonian path, which implies that h(1)=1. Lu et al. (1999) proved the existence of h(k), and showed that h(2)=4 and h(3)=8. The exact values of h(k) remain unknown for k≥4. A result of Erdős on the domination number of tournaments implies h(k)=Ω(klogk). In this paper, we prove that h(4)=10 and h(5)≥13.