On degree sum conditions for directed path-factors with a specified number of paths

Abstract A directed path-factor of a digraph is a spanning subdigraph consisting of a union of vertex-disjoint directed paths in the digraph. In this paper, we give the following result: If D is a digraph of order n ≥ ( 2 l − 1 ) k − 1 , and if d D + ( u ) + d D − ( v ) ≥ n − k for every two distinct vertices u and v with ( u , v ) ∉ A ( D ) , then D has a directed path-factor with exactly k directed paths of order at least l ( ≥ 2 ) . To show this theorem, we discuss the correspondence between digraphs and bipartite graphs with perfect matchings, and also consider degree conditions for the existence of long directed paths in digraphs.