Spanning Components and Surfaces Under Minimum Vertex Degree

TL;DR

This paper determines the minimum vertex-degree thresholds in 3-uniform hypergraphs for the existence of spanning components and surfaces, extending previous work on codegree conditions with asymptotically optimal bounds.

Contribution

It establishes asymptotically tight minimum vertex-degree conditions for spanning components and surfaces in 3-uniform hypergraphs, extending prior codegree results.

Findings

Spanning component exists if minimum vertex degree exceeds half of the possible pairs.

Spanning surface exists if minimum vertex degree exceeds five ninths of the possible pairs.

Results are asymptotically optimal.

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 $\delta_1(G) \gtrsim \tfrac{1}{2} \binom{n}{2}$ and a spanning copy of any surface if $\delta_1(G) \gtrsim \tfrac{5}{9} \binom{n}{2}$, 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.