A Dirac-type theorem for arbitrary Hamiltonian $H$-linked digraphs
Yangyang Cheng, Zhilan Wang, Jin Yan
TL;DR
This work establishes a Dirac-type condition ensuring that a digraph $D$ is arbitrary Hamiltonian $H$-linked for any fixed digraph $H$ with $k$ arcs and $\delta(H)\ge 1$. The authors prove that if $D$ has order $n$ and minimum semi-degree $\delta^0(D)\ge n/2+k$, then every injective vertex mapping of $H$ into $D$ can be extended to a spanning subdivision with prescribed path lengths, using an absorption framework combined with stability analysis. A general form allows $k$ to be linear in $n$ under length restrictions on subdivided arcs, and corollaries recover Wang's conjecture for large graphs and partially address Pavez-Signé's problem. The approach relies on the absorption method, robustness via robust outexpanders, and a detailed extremal-case analysis, advancing the understanding of Hamiltonian linkage in digraphs under Dirac-type conditions.
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 $δ(H)\geq1$, 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-Signé \cite{Pavez}.