Semidegree threshold for spanning trees in oriented graphs

We show that for all $γ> 0$ and $Δ\in \mathbb{N}$, there is some $n_0$ such that, if $n \geq n_0$, then every oriented graph on $n$ vertices with minimum semidegree at least $(3/8 + γ)n$ contains a copy of each oriented tree on $n$ vertices with maximum degree at most $Δ$. This is asymptotically best possible.

💡 Research Summary

This paper determines the asymptotically optimal semidegree condition that guarantees the containment of any bounded‑degree spanning oriented tree in a large oriented graph. The main theorem states that for every fixed γ > 0 and integer Δ, there exists n₀ such that every n‑vertex oriented graph G with minimum semidegree δ⁰(G) ≥ (3/8 + γ)n (and n ≥ n₀) contains a copy of every n‑vertex oriented tree T whose maximum degree Δ(T) ≤ Δ. The bound (3/8 + γ)n matches the known threshold for directed Hamilton cycles and is shown to be asymptotically tight by a construction due to Kelly (based on a 4‑part partition with regular tournaments and a bipartite regular tournament), which yields a graph of semidegree exactly 3n/8 − ½ that fails to contain a directed Hamilton path, and therefore cannot contain certain spanning trees.

The authors’ approach builds on the theory of robust out‑expanders. A digraph D is a robust (ν,τ)‑out‑expander if every vertex set S with τn ≤ |S| ≤ (1 − τ)n has at least |S| + νn vertices receiving at least νn arcs from S. It is known (Kühn‑Osthus‑Treglown) that such digraphs with linear minimum semidegree contain a directed Hamilton cycle. The paper extends this to all bounded‑degree spanning trees (Theorem 1.4). Lemma 1.5 shows that any oriented graph with δ⁰ ≥ (3/8 + γ)n is automatically a robust (ν,τ)‑out‑expander for suitable parameters, thus reducing the main result to proving Theorem 1.4.

The embedding strategy combines several deep tools:

1. Diregularity Lemma – The graph is partitioned into an ε‑regular equipartition {V₀,…,V_k}. The reduced digraph R inherits the semidegree and expansion properties of G.

2. Random walk assignment – Vertices of the tree T are mapped to clusters of R by a random walk that follows the orientation of T. For paths this is straightforward; for general trees a top‑bottom ordering is used, assigning out‑neighbors to a random out‑neighbourhood V⁺ and in‑neighbors to a random in‑neighbourhood V⁻. A martingale analysis guarantees that the walk visits each cluster roughly the same number of times, provided the reduced graph possesses the “cherry property”. The authors prove that any robust expander contains a spanning regular subdigraph with this property.

3. Absorbing property and Blow‑up Lemma – To embed a spanning tree (as opposed to an almost‑spanning one) the authors need the Blow‑up Lemma. They first locate a directed Hamilton cycle C in R (guaranteed by the robust‑expander theorem) and make its arcs super‑regular. Special arcs of T (chosen according to the tree’s structure – many leaves, many bare paths, or many switches) are forced to map onto arcs of C, giving the “absorbing property”. A semi‑random assignment is defined: when the start of a special path is placed, the rest of the path is deterministically embedded along C. Proposition 7.2 shows that this yields both the absorbing property and a balanced distribution of vertices among clusters.

4. Handling exceptional vertices – The regularity partition leaves an exceptional set V₀ (size ≤ εn). Using “skewed‑traverses” (Kelly), each exceptional vertex v is linked to two clusters V_i⁺ and V_i⁻ that receive many arcs from v and send many arcs to v. By reassigning a few vertices of a special path, v can be incorporated without breaking the embedding. Because this technique works only when the path length is a multiple of |V(R)|, the authors split T into two subtrees T_A and T_B and similarly split the host digraph into D_A and D_B, arranging that D_B contains only a tiny exceptional set (size ≤ ξn) which can be handled by skewed‑traverses, while D_A has no exceptional vertices.

5. Perfect balancing – After incorporating all exceptional vertices, a final balancing step (again using skewed‑traverses) adjusts the assignment φ so that each cluster receives exactly the right number of tree vertices, preserving the absorbing property.

With a perfectly balanced assignment in hand, the Blow‑up Lemma is applied to D_B to embed T_B, and similarly to D_A for T_A, finally connecting the two pieces via the pre‑selected special arcs. This completes the embedding of the entire tree T.

The paper concludes with two open problems: (i) determining the exact (non‑asymptotic) semidegree threshold for spanning bounded‑degree trees, and (ii) extending the result to trees whose maximum degree grows with n (e.g., Δ = c n/ log n). The authors note that a random graph with edge‑probability 0.09 has minimum degree ≈0.8 n but typically fails to contain a tree with a star of degree log n/c attached to stars of degree cn/ log n, suggesting that the bound c n/ log n may be essentially optimal up to constants.

Overall, the work provides the first comprehensive Dirac‑type theorem for spanning oriented trees, aligning the semidegree threshold with that for directed Hamilton cycles, and showcases a sophisticated combination of robust expansion, regularity, random walks, and absorption techniques that will likely influence future research on embedding problems in directed and oriented graphs.