Paths partition with prescribed beginnings in digraphs: A Chvátal–Erdős condition approach

Abstract

A digraph D verifies the Chvátal–Erdős conditions if α ( D ) ⩽ κ ( D ) , where α ( D ) is the stability number of D and κ ( D ) 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 α = α ( D ) vertices { x 1 , … , x α } , there exists a vertex-partition of D into directed paths { P 1 , … , P α } such that P i begins at x i for all i. The case α ( D ) = 2 of the conjecture is proved.