Paths and Stability Number in Digraphs
The Gallai-Milgram theorem says that the vertex set of any digraph with stability number $k$ can be partitioned into $k$ directed paths. In 1990, Hahn and Jackson conjectured that this theorem is best possible in the following strong sense. For each positive integer $k$, there is a digraph $D$ with stability number $k$ such that deleting the vertices of any $k-1$ directed paths in $D$ leaves a digraph with stability number $k$. In this note, we prove this conjecture.