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

TL;DR

This paper establishes minimum semi-degree conditions for large oriented graphs to be arbitrary H-linked, extending Hamiltonicity results and providing tight bounds for cycle-factors and q-linked properties.

Contribution

It introduces a semi-degree threshold ensuring large oriented graphs are arbitrary H-linked, generalizing previous Hamilton cycle and cycle-factor results.

Findings

Determines semi-degree bounds for arbitrary H-linked oriented graphs.

Extends Hamiltonicity results to arbitrary H-linked graphs.

Provides tight bounds for strongly Hamiltonian-connected and q-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, K\"{u}hn, and Osthus proved that every sufficiently large oriented graph $ D $ of order $ n $ with $\delta^0(D) \geq \frac{3n-4}{8}$ contains a Hamilton cycle (i.e., a $\overset{\leftrightarrow}{K_2}$-subdivision). Subsequently, Kelly, K\"{u}hn, 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 \geq n_0 $ with $\delta^0(D) \geq \frac{3n + 3h + 3q - 5}{8}$ is arbitrary $ H $-linked; specifically, if $H$ is a loop, this holds under the weaker condition $\delta^0(D) \geq \frac{3n - 4}{8}$. 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.