Degree conditions for $k$-strong orientations of digraphs

Jackson and Thomassen conjectured that every $2k$-strong digraph contains a spanning $k$-strong oriented subdigraph. We prove sharp degree conditions for the existence of such a subdigraph. For every fixed positive integer $k$ and all sufficiently large $n$, every $n$-vertex digraph $D$ with ${\delta }^0(D)\ge \lfloor (n+k-1)\mathrm{/}2\rfloor$ admits a $k$-strong orientation. This threshold is best possible even for the weaker conclusion that $D$ itself is $k$-strong. We also prove a sharp Woodall-type analogue for every fixed positive integer $k$ and all sufficiently large $n$: if $d_D^{+}(x)+d_D^{-}(y)\ge n+2k-2$ for every missing arc $xy$, then $D$ admits a $k$-strong orientation, and this bound is again best possible. As a further consequence, we determine the sharp minimum total degree threshold. Finally, the semi-degree result also remains valid when $k\le \alpha n$ for every fixed $0<\alpha <0.094882\dots$.