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arXiv · 2407.06275

Spanning spheres in Dirac hypergraphs

Freddie Illingworth·Richard Lang·Alp Müyesser·Olaf Parczyk·Amedeo Sgueglia
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Abstract

We show that a $k$-uniform hypergraph on $n$ vertices has a spanning subgraph homeomorphic to the $(k - 1)$-dimensional sphere provided that $H$ has no isolated vertices and each set of $k - 1$ vertices supported by an edge is contained in at least $n/2 + o(n)$ edges. This gives a topological extension of Dirac's theorem and asymptotically confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan. Unlike typical results in the area, our proof does not rely on the Absorption Method, the Regularity Lemma or the Blow-up Lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host graph with a family of complete blow-ups.

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Freddie Illingworth, Richard Lang, Alp Müyesser, Olaf Parczyk, Amedeo Sgueglia. 2024-07-08. Spanning spheres in Dirac hypergraphs. https://arxiv.org/abs/2407.06275

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