Blowing up Dirac's theorem

TL;DR

This paper proves that graphs with a minimum degree slightly above half of the number of vertices contain a large, nearly uniform cycle blow-up, expanding Dirac's theorem with a new blow-up cover approach.

Contribution

It introduces a novel method using blow-up covers to find spanning substructures, extending Dirac's theorem to cycle blow-ups in dense graphs.

Findings

Graphs with minimum degree > (1/2 + ε)n contain cycle blow-ups.

The cycle blow-up spans the entire graph.

Clusters are of size Ω(log n).

Abstract

We show that every graph $G$ on $n$ vertices with $\delta(G) \geq (1/2+\varepsilon)n$ is spanned by a complete blow-up of a cycle with clusters of nearly uniform size $\Omega(\log n)$. The proof is based on a recently introduced approach for finding vertex-spanning substructures via blow-up covers.