Degree conditions for disjoint path covers in digraphs

In this paper, we study degree conditions for three types of disjoint directed path cover problems: the many-to-many k-DDPC problem, the one-to-many k-DDPC problem and the one-to-one k-DDPC problem. Firstly, we get two sharp minimum semi-degree sufficient conditions for the unpaired many-to-many k-DDPC problem and a sharp Ore-type degree condition for the paired many-to-many $2$-DDPC problem. Secondly, we obtain a minimum semi-degree sufficient condition for the one-to-many k-DDPC problem on a digraph with order n, and show that the bound is sharp when $n+k$ is even and is sharp up to an additive constant 1 otherwise. Finally, we give a minimum semi-degree sufficient condition for the one-to-one k-DDPC problem on a digraph with order n, and show that the bound is sharp when $n+k$ is odd and is sharp up to an additive constant 1 otherwise.