[Paper Review] Turán-type and tiling problems in oriented graphs

Ming Chen, Wenxu Lu+2 more|arXiv (Cornell University)|Mar 23, 2026

Limits and Structures in Graph Theory0 citations

TL;DR

The paper extends Turán-type and tiling results from C3 tilings to D_{a,b,c}-tilings in large oriented graphs with minimum semi-degree about n/2, and proves near-perfect tilings with stability; it also provides lower bounds for semi-degree thresholds of cycle and path powers.

ABSTRACT

Given $a,b,c\in\mathbb N$, let $D_{a,b,c}$ be the tournament on $a+b+c$ vertices obtained by replacing the vertices of the directed triangle $C_3$ with transitive tournaments $TT_a$, $TT_b$, and $TT_c$, respectively. Keevash and Sudakov (2009) showed that every sufficiently large oriented graph $G$ on $n$ vertices with $δ^{0}(G)\geqslant (1/2-o(1))n$ contains a $C_3$-tiling, equivalently a $D_{1,1,1}$-tiling, covering all but at most three vertices. We generalize this result to arbitrary blow-ups $D_{a,b,c}$. Specifically, for any fixed $a,b,c$, every sufficiently large oriented graph $G$ on $n$ vertices with $δ^{0}(G)\geqslant (1/2-o(1))n$ contains a $D_{a,b,c}$-tiling covering all but at most $2(a+b+c)-3$ vertices. Moreover, this bound is essentially sharp. We also establish a stronger stability result: if $(a+b+c)\mid n$, then either $G$ contains a $D_{a,b,c}$-factor, or $G$ is close to an extremal graph. Our interest in $D_{a,b,c}$ is also motivated by oriented Turán theory: a seminal theorem of Bollobás and Häggkvist (1990) shows that a tournament $T$ is Turánable (i.e., contained in every sufficiently large regular tournament) if and only if $T\subseteq D_{s,s,s}$ for some $s$. Complementing our tiling results, we also investigate related semi-degree thresholds for powers of directed cycles and paths. In particular, we present two $n$-vertex constructions that give lower bounds, showing that the minimum semi-degree thresholds for $C^2_l$ with $l ot\equiv 0\pmod 6$ and for $P^2_l$ with $l\geqslant 7$ are at least $4n/9$ and $3n/8$, respectively.

Motivation & Objective

• Generalize C3-tiling results to D_{a,b,c}-tilings and -factors in large oriented graphs under near half-regularity conditions.
• Establish near-perfect tilings: every large oriented graph with delta^0(G) >= (1/2 - o(1))n contains a D_{a,b,c}-tiling covering all but 2(a+b+c)-3 vertices.
• Prove a stronger stability result: under divisibility, either a D_{a,b,c}-factor exists or G is close to an extremal structure.
• Investigate semi-degree thresholds for powers of directed cycles and paths, providing new lower bound constructions.

Proposed method

• Define and study the D_{a,b,c} family as blow-ups of the directed triangle with transitive tournaments TT_a, TT_b, TT_c.
• Use absorption method to obtain an absorbing set for D_{a,b,c}-factors (Lemmas 5.2–5.3).
• Apply the Regularity Lemma for digraphs (Diregularity Lemma) and reduced digraph techniques to embed D_{a,b,c} in G.
• Develop a framework of (H,β,t)-closed vertex sets and linking techniques to achieve almost coverings and extend to full tilings.
• Establish extremal and stability structures (γ-extremal and γ-superextremal partitions) to handle near-extremal cases.
• Rely on congruence and gcd-type conditions to determine when D_{a,b,c}-factors are guaranteed (Theorems 1.4 and 1.5).

Experimental results

Research questions

• RQ1 What minimum semi-degree is necessary (asymptotically) to guarantee a D_{a,b,c}-tiling or -factor in large oriented graphs?
• RQ2 Under what gcd and divisibility conditions does a D_{a,b,c}-factor exist in large semi-regular tournaments?
• RQ3 How close can a graph be to extremal structure if it fails to contain a D_{a,b,c}-factor under near-regular conditions?
• RQ4 What are the lower bounds for semi-degree thresholds for powers of directed cycles and paths, and how do they compare to known conjectures?
• RQ5 To what extent do the D_{a,b,c} tilings bridge cycle-tiling and transitive-tournament tiling problems?

Key findings

• For any fixed a,b,c, every sufficiently large oriented graph on n vertices with delta^0(G) >= (1/2 - o(1))n contains a D_{a,b,c}-tiling covering all but at most 2(a+b+c)-3 vertices.
• If (a+b+c) divides n, then either G contains a D_{a,b,c}-factor or G is close to an extremal γ-extremal structure.
• Two gcd-based conditions ensure existence of D_{a,b,c}-factors in large semi-regular tournaments: gcd(a+b+c,c^2 - ab) = 1 (for general a,b,c with a≥2) and gcd(1+b+c,c^2 - b) = 1 (for a=1).
• Propositions show there exist semi-regular tournaments that do not contain D_{a,b,c}-factors when gcd(a+b+c,c^2 - ab) > 1 (and analogously for D_{1,b,c} when gcd(1+b+c,c^2 - b) > 1).
• The paper provides lower bound constructions yielding kappa^0(C_l^2) >= 4/9 for l not multiple of 6 and kappa^0(P_l^2) >= 3/8 for l>=7.
• An absorbing-lemma-based approach yields an Almost Covering Lemma: under near-regular degree, a D_{a,b,c}-tiling covers all but a small fraction, which can be completed to a factor in the final step.