Spanning Components and Surfaces Under Minimum Vertex Degree

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 $\hat{I}{´}_1(G)\gtrsim \frac{1}{2}\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ and a spanning copy of any surface if $\hat{I}{´}_1(G)\gtrsim \frac{5}{9}\left(\genfrac{}{}{0ex}{}{n}{2}\right)$, 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.

15 pages, 2 figures