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 ( 2 k − 2 ) -strong semicomplete digraphs which contain no spanning k -strong tournament and conjecture that every ( 2 k − 1 ) -strong semicomplete digraph which is not the complete digraph K 2 k ∗ on 2 k vertices contains a spanning k -strong tournament.