Spanning Components and Surfaces Under Minimum Vertex Degree

TL;DR

The paper advances Dirac-type results for 3-uniform hypergraphs by establishing asymptotically optimal minimum vertex-degree thresholds for spanning components and spanning surfaces: $\delta_1(G) \ge (1/2+o(1))\binom{n}{2}$ ensures a spanning component, and $\delta_1(G) \ge (5/9+o(1))\binom{n}{2}$ ensures a spanning copy of any surface. It introduces the Hamilton framework machinery, showing that large minimum degrees enforce a subgraph with connectivity, space, and aperiodicity that yields Hamiltonicity, and uses blow-ups of cycles to embed spanning spheres and surfaces. The work also provides tight lower-bound constructions demonstrating asymptotic optimality and clarifies the relationship between spanning structures and Hamiltonicity in the 3-graph setting, including a near-coincidence of thresholds for spheres and Hamilton frameworks. Overall, the study delivers a unified embedding framework for spanning geometric-like structures in 3-uniform hypergraphs and opens avenues for higher-uniformity generalizations and deeper connections to Hamiltonicity.

Abstract

We study minimum vertex-degree conditions in 3-uniform hypergraphs for (tight) spanning components and (combinatorial) surfaces. Our main results show that a 3-uniform hypergraph $G$ on $n$ vertices contains a spanning component if $δ_1(G) \gtrsim \tfrac{1}{2} \binom{n}{2}$ and a spanning copy of any surface if $δ_1(G) \gtrsim \tfrac{5}{9} \binom{n}{2}$, which in both cases is asymptotically optimal. This extends the work of Georgakopoulos, Haslegrave, Montgomery, and Narayanan who determined the corresponding minimum codegree conditions in this setting.

Figures (2)

• Figure 1: A $3$-graph without a spanning component. The triangles indicate which $3$-sets are edges. The edges corresponding to the dashed and non-dashed triangles each form a component.
• Figure 2: The edge sets of $\mathcal{T}_{9}$ and $\mathcal{P}_{12}$ consist of all the triangles in the respective figures.

Theorems & Definitions (29)

• Lemma 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 1.6
• Theorem 1.7
• proof : Proof of \ref{['lem:HamCompDegOptimal']}
• proof : Proof of \ref{['thm:spanning-component']}
• Claim 2.1