Spanning surfaces in 3-graphs

Agelos Georgakopoulos,Bhargav Narayanan,John Haslegrave,Richard Montgomery

doi:10.4171/jems/1101

Abstract

We prove a topological extension of Dirac's theorem suggested by Gowers in 2005: for any connected, closed surface $\mathscr{S}$, we show that any two-dimensional simplicial complex on $n$ vertices in which each pair of vertices belongs to at least $n/3 + o(n)$ facets contains a homeomorph of $\mathscr{S}$ spanning all the vertices. This result is asymptotically sharp, and implies in particular that any 3-uniform hypergraph on $n$ vertices with minimum codegree exceeding $n/3+o(n)$ contains a spanning triangulation of the $2$-sphere.

Highlights

We extend the classical graph-theoretic result of Dirac [4] on spanning cycles to the setting of simplicial 2-complexes, or equivalently, the setting of 3-uniform hypergraphs
A natural generalisation is to treat a ‘spanning cycle in a 3-graph’ as a triangulation of the 2-sphere spanning the vertex set, and here we determine asymptotically the best-possible minimum codegree condition which guarantees the existence of such an object in an n-vertex 3-graph
Appealing to Lemma 4.13, we find a collection S1 of at most ηn vertex-disjoint green-tinged spheres disjoint from R, which can absorb any subset of R and which have at most μn/9 vertices in total; let U be the set of vertices belonging to spheres in S1

Summary

Introduction

We extend the classical graph-theoretic result of Dirac [4] on spanning cycles to the setting of simplicial 2-complexes, or equivalently, the setting of 3-uniform hypergraphs (or 3-graphs for short). Problem 1.1 asks for degree conditions that guarantee the existence of a homeomorphic copy of S2 containing all the vertices in a simplicial 2-complex. As we shall see (in Proposition 3.1), a 3-graph H on n vertices whose minimum codegree exceeds n/3 contains a spanning tight component It may have another non-spanning tight component whose edges are of no use whatsoever when trying to build a spanning copy of any surface. It has been brought to our attention that Conlon, Ellis and Keevash [3] earlier proved (in unpublished work, as previously mentioned) a weaker statement in the direction of Theorem 1.4, showing that any 3-graph H on n vertices with δ2(H) ≥ 2n/3 + o(n) contains a spanning copy of the sphere.

Preliminaries

Overview of our strategy

Proof of the main result

Conclusion