A digraph D verifies the Chvátal–Erdős conditions if , where is the stability number of D and is its vertex-connectivity. Related to the Gallai–Milgram Theorem (see Gallai and Milgram [Verallgemeinerung eines Graphentheorischen Satzes von Redei, Acta Sci. Math. 21 (1960) 181–186]), we raise in this context the following conjecture. For every set of vertices , there exists a vertex-partition of D into directed paths such that begins at for all i. The case of the conjecture is proved.