Highly connected spanning oriented subdigraphs in generalizations of semicomplete digraphs

Let $k$ be a positive integer. Jackson and Thomassen conjectured in 1989 that there exists an integer function $f(k)$ such that every $f(k)$-strong digraph admits a spanning $k$-strong oriented subdigraph. They even conjectured that one can take $f(k)=2k$ [Ann. N. Y. Acad. Sci. 555 (1989) 402-412]. Already the existence of $f(2)$ is open for general digraphs. Thomassen proved that $f(2)=4$ for symmetric digraphs. For general $k$, the existence of $f(k)$ was only known for locally semicomplete digraphs and quasi-transitive digraphs. Guo proved that every ${(3k-2)}$-strong locally semicomplete digraph contains a spanning $k$-strong local tournament [Discrete Appl. Math. 79 (1997) 119--125]. One can deduce from Guo's result that we have $f(k)\leq 3k-2$ for quasi-transitive digraphs. In this paper, we prove the existence of $f(k)$ for two subclasses of the semicomplete multipartite digraphs, namely extended semicomplete digraphs and semicomplete split digraphs. We prove that every $(4k+1)$-strong extended semicomplete digraph contains a spanning $k$-strong oriented subdigraph and every $5k$-strong semicomplete split digraph contains a spanning $k$-strong oriented subdigraph. The first result implies that for the large class of digraphs which can be obtained from some semicomplete digraph $S$ on at least 3 vertices by substituting arbitrary digraphs for each vertex of $S$ we also have $f(k)\leq 4k+1$.