Degree conditions for disjoint path covers in digraphs

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

A Dirac-type theorem for arbitrary Hamiltonian HH-linked digraphs

Given any digraph $D$, let $\mathcal{P}(D)$ be the family of all directed paths in $D$, and let $H$ be a digraph with the arc set $A(H)=\{a_1, \ldots, a_k\}$. The digraph $D$ is called arbitrary Hamiltonian $H$-linked if for any injective mapping $f: V(H)\rightarrow V(D)$ and any integer set $\mathcal{N}=\{n_1, \ldots, n_k\}$ with $n_i\geq4$ for each $i\in\{1, \ldots, k\}$, there exists a mapping $g: A(H)\rightarrow \mathcal{P}(D)$ such that for every arc $a_i=uv$, $g(a_i)$ is a directed path from $f(u)$ to $f(v)$ of length $n_i$, and different arcs are mapped into internally vertex-disjoint directed paths in $D$, and $\bigcup_{i\in[k]}V(g(a_i))=V(D)$. In this paper, we prove that for any digraph $H$ with $k$ arcs and $\delta(H)\geq1$, every digraph of sufficiently large order $n$ with minimum in- and out-degree at least $n/2+k$ is arbitrary Hamiltonian $H$-linked. Furthermore, we show that the lower bound is best possible. Our main result extends some work of K\"{u}hn and Osthus et al. \cite{20081,20082} and Ferrara, Jacobson and Pfender \cite{Jacobson}. Besides, as a corollary of our main theorem, we solve a conjecture of Wang \cite{Wang} for sufficiently large graphs