Abstract
The Gallai-Milgram theorem asserts that the vertex set of any digraph with stability number $k$ can be partitioned into $k$ directed paths. Hahn and Jackson conjectured that, for any positive integer $k$, there exists a digraph with stability number $k$ such that the subdigraph obtained by deleting any $k-1$ directed paths still has stability number $k$. They established the existence of such digraphs for $k=1,2,3$. Here we construct examples for arbitrarily large values of $k$.
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