Semi-Degree Condition for Arbitrary $H$-Linked Oriented Graphs

Abstract

Let $H$ be a multi-digraph on $h$ vertices with $q$ arcs. An \textbf{$H$-subdivision} in a digraph $D$ is a subdigraph obtained by replacing every arc $uv$ of $H$ with a path from $u$ to $v$ in $D$ such that these paths are pairwise internally vertex-disjoint. A digraph $D$ is \textbf{arbitrary $H$-linked} if, for every injection $f:V(H)\to V(D)$, there exists an $H$-subdivision in $D$ such that each vertex $v\in V(H)$ is mapped to $f(v)\in V(D)$, and the length of every subdivision path can be arbitrarily specified as {an integer \(l \geq 4\)}. An oriented graph is a digraph without 2-cycles. Keevash, Kühn, and Osthus proved that every sufficiently large oriented graph $D$ of order $n$ with $\hat{I}{´}^0(D)\ge \frac{3n-4}{8}$ contains a Hamilton cycle (i.e., a $\overleftrightarrow{K_2}$-subdivision). Subsequently, Kelly, Kühn, and Osthus showed that such oriented graphs {are also arbitrary $H$-linked, where $H$ is a loop}. Motivated by these results, we establish a minimum semi-degree condition for arbitrary $H$-linked oriented graphs: there exists $n_0=n_0(h,q)$ such that every oriented graph $D$ of order $n\ge n_0$ with $\hat{I}{´}^0(D)\ge \frac{3n+3h+3q-5}{8}$ is arbitrary $H$-linked; specifically, if $H$ is a loop, this holds under the weaker condition $\hat{I}{´}^0(D)\ge \frac{3n-4}{8}$. The result provides an oriented graph analogue of Wang's conjecture on cycle-factors in graphs [J. Korean Math. Soc. 51 (2014) 919--940] and determines the tight semi-degree bounds for both strongly Hamiltonian-connected and arbitrary $q$-linked oriented graphs.

39 pages,6 figures