Blowing up Dirac's theorem

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