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 linkage.

We first 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. We then 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. Furthermore, these results hold even when n is (at least) a linear function of k. In addition, our results improve the existing results by reducing both of the lower bounds of the order and the minimum semi-degree condition of digraphs.