Jackson–Thomassen conjecture

Posed by Jackson and Thomassen in 1989, the conjecture says that every $2k$-strong digraph has a spanning $k$-strong oriented subdigraph. It is known for $k=1$, but remains open for general digraphs already at $k=2$.

September 2026 partial results

• Bang-Jensen and Wei report sharp sufficient degree conditions, including $\delta^0(D)\ge\lfloor(n+k-1)/2\rfloor$ for fixed $k$ and sufficiently large $n$, plus an extension to $k\le\alpha n$ with $\alpha<0.094882\ldots$.
• Other September papers establish the conjectured conclusion for extended semicomplete, semicomplete split, and related composition classes, but not arbitrary digraphs.
• A separate result settles the analogous arc-connectivity statement, not the vertex-connectivity conjecture.

Current status (as of September 2026): $k=1$ and several structured or dense cases are known, but the general Jackson–Thomassen conjecture remains open, including general $k=2$.