On the bounded domination number of tournaments

In a simple digraph, a star of degree t is a union of t edges with a common tail. The k-domination number γk(G) of digraph G is the minimum number of stars of degree at most k needed to cover the vertex set. We prove that γk(T)=⌈n/(k+1)⌉ when T is a tournament with n⩾14k lg k vertices. This improves a result of Chen, Lu and West. We also give a short direct proof of the result of E. Szekeres and G. Szekeres that every n-vertex tournament is dominated by at most lg n−lg lg n+2 vertices.