Degree conditions for disjoint path covers in digraphs

Abstract: In this paper, we study degree conditions for three types of disjoint directed path cover problems: many-to-many $k$-DDPC, one-to-many $k$-DDPC and one-to-one $k$-DDPC, which are intimately connected to other famous topics in graph theory, such as Hamiltonicity and $k$-linkage, and have a strong background of applications. 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 for the minimum semi-degree 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 for the minimum semi-degree is sharp when $n+k$ is odd and is sharp up to an additive constant 1 otherwise.