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

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 >= (2l - 1)k - 1, and if d(D)(+)(u)+ d(D)(-) (v) >= n - k for every two distinct vertices u and v with (u, v) is not an element of 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. (c) 2020 Elsevier B.V. All rights reserved.