3-Uniform Hypergraphs

• 3-uniform hypergraphs are defined as hypergraphs where every edge is a 3-element subset, forming the basis for studies in Hamilton cycles and tilings.
• Key results establish sharp vertex-degree thresholds for loose and tight Hamilton cycles using techniques like the absorption and reservoir methods.
• Research applications include perfect matchings, spanning surface embeddings, and tilings, with extremal constructions demonstrating the sharpness of these thresholds.

A 3-uniform hypergraph (3-graph) is a hypergraph $H = (V,E)$ in which every edge $e \in E$ is a 3-element subset of the finite vertex set $V$. The study of minimum vertex degree and higher-order degree constraints in 3-uniform hypergraphs has yielded sharp thresholds for spanning structures, various tilings, and connections with classical extremal combinatorics. This entry surveys central results, techniques, and open questions in the theory of 3-uniform hypergraphs, highlighting influential theorems for Hamiltonicity, perfect matchings, tilings, and topological embeddings.

1. Structure and Definitions

A 3-uniform hypergraph is defined as a pair $H = (V,E)$, where $V$ is a finite set of vertices and $E \subseteq \binom{V}{3}$ is a collection of 3-element subsets of $V$. The vertex-degree of $v \in V$ is $\deg_H(v) = |\{e \in E : v \in e\}|$, and the minimum vertex degree is

$\delta_1(H) = \min_{v \in V} \deg_H(v).$

For $d\geq1$, the minimum $d$-degree $\delta_d(H)$ is the minimal number of edges containing any fixed $d$-subset of vertices.

Substructures of interest include cycles, tilings (packings), spanning components, and combinatorial embeddings of topological surfaces.

• Loose cycle: a sequence of edges $e_1, \dots, e_t$ such that each consecutive pair $e_i, e_{i+1}$ shares exactly one vertex and all other intersections are empty.
• Tight cycle: an ordering $v_1, \dots, v_n$ of $V$ so that every edge is $\{v_i, v_{i+1}, v_{i+2}\}$ (indices modulo $n$), i.e., consecutive edges overlap in exactly two vertices.
• Tilings: collections of vertex-disjoint copies of a fixed subhypergraph covering all vertices (perfect tiling/factor).
• Tight component: a subhypergraph whose line graph (edges as vertices, adjacency by intersection in 2 vertices) is connected; spanning if it includes all vertices.

2. Hamiltonicity: Loose and Tight Hamilton Cycles

Loose Hamilton Cycles

A loose Hamilton cycle is a spanning loose cycle: a cyclic ordering of the $n$ vertices ($n$ even) so that the edges are every three consecutive vertices, and consecutive edges intersect in exactly one vertex. The sharp minimum vertex-degree threshold is: $$\delta_1(H) \geq \binom{n-1}{2} - \binom{\lfloor 3n/4 \rfloor}{2} + c$$ where $c = 2$ for $n \equiv 0 \pmod{4}$ and $c = 1$ for $n \equiv 2 \pmod{4}$, for all even $n > n_0$ (Han et al., 2013). This is best possible, with extremal configurations obtained by partitioning $V = A \dot{\cup} B$, taking all edges meeting $A$, and $|A| \approx n/4$, so that $B$ is independent and no loose Hamilton cycle can exist.

Asymptotically, this matches the bound

$$\delta_1(H) \ge \left( \frac{7}{16} + o(1) \right) \binom{n}{2}$$

for large $n$, as proved by Buß–Hàn–Schacht (Buß et al., 2016).

The proofs are based on the absorbing method, combining construction of a short absorbing path, a small reservoir, almost perfect tilings by loose paths, and a final absorption argument.

Tight Hamilton Cycles

For tight Hamilton cycles, the sharp asymptotic minimum vertex-degree condition is (Reiher et al., 2016): $$\delta_1(H) \ge \left( \frac{5}{9} + o(1) \right) \binom{n}{2}.$$ Here, the threshold is witnessed by constructions partitioning $V = X \cup Y$, taking all edges meeting $X$ in at most one vertex (for $|X| \sim n/3$) or at most two vertices (for $|X| \sim 2n/3$), which have the required minimum vertex-degree but do not support tight Hamilton cycles.

This threshold arises from the absorption method tailored to tight cycles, using robust link-graphs, connectability notions, a random reservoir, and precise combinatorial absorption structures.

3. Spanning Components and Topological Embeddings

The minimum vertex-degree threshold for the existence of a spanning tight component in a 3-uniform hypergraph is (Allsop et al., 30 Dec 2025): $\delta_1(G) \ge \frac{1}{2} \binom{n-1}{2} + 1$ which is both necessary and sufficient up to lower-order terms. The constructive obstruction partitions $V$ into $X, Y, Z$ with $|Z|=1, |X| \sim n/\sqrt{2}$, and takes only certain classes of edges, resulting in two tight components, neither spanning.

For embeddings of closed connected 2-dimensional surfaces as spanning triangulations (inducing a homeomorphic 2-complex), the threshold is asymptotically

$$\delta_1(G) \ge \left( \frac{5}{9} + o(1) \right) \binom{n}{2}$$

which is again optimal. The proof combines a Hamilton framework argument and combinatorial-topological constructions, showing that for any surface $S$ and sufficiently large $n$, the minimum vertex-degree above this threshold guarantees a spanning triangulation of $S$.

4. Perfect Tilings and Packings

The minimum vertex-degree thresholds for perfect tilings in 3-uniform hypergraphs depend on the structure of the target subgraph.

Cyclic Tilings

For perfect tilings of the 3-uniform cycle on four vertices with two edges ($C_4^3$), the sharp threshold for $n \in 4 \mathbb{N}$ is (Han et al., 2013): $$\delta_1(H) \geq \binom{n-1}{2} - \binom{3n/4}{2} + \frac{3}{8} n + c(n)$$ where $c(n) = 1$ for $n \in 8 \mathbb{N}$, $c(n) = -1/2$ otherwise.

Complete Multipartite Factors

For a complete 3-partite 3-uniform hypergraph $K_{a,b,c}$ with $a \le b \le c$, set $k = a+b+c$. The minimum vertex-degree threshold for a $K_{a,b,c}$-factor is (Han et al., 2015): $$t_1\left(n, K_{a,b,c}\right) = \left( \alpha(a,b,c) + o(1) \right) \binom{n}{2}$$ where $\alpha(a,b,c)$ is the maximum of several explicit terms capturing space and divisibility barriers related to extremal constructions.

The proof utilizes a lattice-based absorbing method, weak regularity lemma, and fractional homomorphism tilings.

5. Proof Techniques

The absorption method is ubiquitous in the sharp results for Hamiltonicity and perfect tilings. Its key ingredients include:

• Absorbing Lemma: Construction of small structures (absorbers) that can incorporate leftover vertices into the desired subgraph or cycle.
• Reservoir Lemma: Selection of a small subset of vertices (the reservoir) that allows for flexible connections between different substructures.
• Path/Cycle-Tiling Lemma: Use of hypergraph regularity and embedding tools to cover most vertices by the desired configuration, typically leaving only a small set for absorption.

For tight connectivity and topological embeddings, combinatorial properties of link graphs and Hamilton frameworks are instrumental, along with precise intersection counting arguments.

6. Extremal Constructions and Sharpness

The sharpness of each threshold is demonstrated by extremal examples:

• For loose Hamiltonicity and $C_4^3$-tiling, the key constructions involve partitioning $V$ such that a large subset is independent or supports only partial packings.
• For $K_{a,b,c}$-factors, structured partitions and divisibility obstructions, or constraints on degrees/coverings, provide the sharp barriers.
• For spanning surfaces, the construction partitions $V$ such that topological constraints (e.g., Euler’s formula) cannot be satisfied in any spanning triangulation, even when the minimum vertex-degree is just below threshold.

7. Open Problems and Directions

Open questions include:

• Pinning down exact (not just asymptotic) vertex-degree thresholds for more general classes of tilings, higher uniformities, and additional substructures.
• Determining vertex-degree thresholds for perfect matchings in 3-graphs (the analog of the Rödl–Ruciński perfect matching conjecture for codegree).
• Extension of absorption and stability methods to hypergraph Turán-type and stability problems; addressing divisibility and fractional tiling obstructions.
• Investigating the interplay between minimum codegree and vertex-degree thresholds, especially for surfaces and higher-dimensional combinatorial embeddings.
• Tight thresholds for sparse regimes and for generalizations beyond 3-uniform hypergraphs to $k$-uniform hypergraphs, where $k > 3$.

The survey of 3-uniform hypergraph theory thus exposes a rich structure of phase transitions, optimal extremal configurations, and intricate combinatorial proof methods, with deep connections to both discrete mathematics and topological combinatorics (Han et al., 2013, Buß et al., 2016, Allsop et al., 30 Dec 2025, Han et al., 2013, Han et al., 2015, Reiher et al., 2016).