A Dirac-type theorem for arbitrary Hamiltonian $H$-linked digraphs

Abstract

Given any digraph $D$ on $n$ vertices, 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 map $f: V(H)\rightarrow V(D)$ and any integer set $\mathcal{N}=\{n_1, \ldots, n_k\}$ satisfying that $n_i\geq4$ for each $i\in\{1, \ldots, k\}$, there is a map $g: A(H)\rightarrow \mathcal{P}(D)$ 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 $\bigcup_{i\in[k]}V(g(a_i))=V(D)$. Here, the length of a directed path is defined as the number of its arcs. In this paper, we prove that for any digraph $H$ with $k$ arcs and $\delta(H)\geq1$, there exists a constant $C_0=C_0(k)$ such that if $D$ is a digraph of order $n\geq C_0$ and minimum in- and out-degree at least $n/2+k$, then it is arbitrary Hamiltonian $H$-linked. The lower bound on the minimum in- and out-degree is best possible. We further prove a more general form that allows $k$ to be linear in $n$, while imposing some restrictions on the lengths of the subdivided arcs. As corollaries, we solved a conjecture of Wang \cite{Wang} for sufficiently large graphs, and partly answered a problem raised by Pavez-Sign\'{e} \cite{Pavez}.