Spanning 2-strong tournaments in 3-strong semicomplete digraphs

We prove that every 3-strong semicomplete digraph on at least 5 vertices contains a spanning 2-strong tournament. Our proof is constructive and implies a polynomial algorithm for finding a spanning 2-strong tournament in a given 3-strong semicomplete digraph. We also show that there are infinitely many (2k−2)(2k−2)-strong semicomplete digraphs which contain no spanning kk-strong tournament and conjecture that every(2k−1)(2k−1)-strong semicomplete digraph which is not the complete digraph K2k∗ on 2k2k vertices contains a spanning kk-strong tournament.