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

TL;DR

This work establishes a tight semi-degree threshold for oriented graphs to be arbitrarily H-linked for any multi-digraph H with h vertices and q arcs. The authors prove that, for sufficiently large n, δ^0(D) ≥ (3n+3h+3q−5)/8 guarantees arbitrary H-linkage, with a weaker bound when H is a loop. The proof blends extremal partitioning, short- and long-path construction, probabilistic multi-partitioning, and pinch-vertex techniques to reduce long subdivisions to Hamilton cycles in auxiliary graphs, leveraging Keevash–Kühn–Osthus theory. The results connect to cycle-factor analogues and Wang’s conjecture, and they establish tight bounds for both strongly Hamiltonian-connected and arbitrary 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, Kühn, and Osthus proved that every sufficiently large oriented graph $ D $ of order $ n $ with $δ^0(D) \geq \frac{3n-4}{8}$ contains a Hamilton cycle (i.e., a $\overset{\leftrightarrow}{K_2}$-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 \geq n_0 $ with $δ^0(D) \geq \frac{3n + 3h + 3q - 5}{8}$ is arbitrary $ H $-linked; specifically, if $H$ is a loop, this holds under the weaker condition $δ^0(D) \geq \frac{3n - 4}{8}$. 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.

Figures (5)

• Figure 1: An oriented graph $D$ on $n$ vertices with $\delta^0(D)\geq (3n+3h+3q-13)/8$ that is not arbitrary Hamiltonian $H$-linked.
• Figure 2: An oriented graph $D$ on $n$ vertices with $\delta^0(D)\geq 13n/32 - 3$ that contains no $(u,v)$-path with length 3.
• Figure 3: An $xy$-butterfly.
• Figure 4: An extremal partition $(D_1, D_2, D_3, D_4)$.
• Figure 5: Illustration of $H'$: Solid lines denote arcs in $H'$ (if such arcs exist in $D$), while dashed lines denote arcs that lie in the digraph $D$ but not in $H'$; the paths formed by the concatenation of solid and dashed lines are exactly the paths $\{Q'_i \mid i\in [r]\}$ we seek.

Theorems & Definitions (41)

• Theorem 1.1
• Corollary 1.1
• Conjecture 1.1
• Proposition 2.1
• Proof 1
• Proposition 2.2
• Proof 2
• Lemma 3.1
• Proof 3
• Lemma 3.2