Regular tournament ($d^{+}(v)=d^{-}(v)=(N-1)/2$)

A regular tournament has equal outdegree and indegree at every vertex. On $2k+1$ vertices, label by residues modulo $2k+1$ and direct $i\to j$ when $j-i\in \{1,\dots ,k\}$. This gives an explicit regular tournament. It contains no outward-oriented star with more than $k$ leaves.