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We consider sufficient conditions for the existence of $k$-th powers ofHamiltonian cycles in $n$-vertex graphs $G$ with minimum degree $\mu n$ forarbitrarily small $\mu>0$. About 20 years ago Koml\'os, Sark\"ozy, andSzemer\'edi resolved the conjectures of P\'osa and Seymour and obtained optimalminimum degree conditions for this problem by showing that $\mu=\frac{k}{k+1}$suffices for large $n$. For smaller values of $\mu$ the given graph $G$ mustsatisfy additional assumptions. We show that inducing subgraphs of density$d>0$ on linear subsets of vertices and being inseparable, in the sense thatevery cut has density at least $\mu>0$, are sufficient assumptions for thisproblem and, in fact, for a variant of the bandwidth theorem. This generalisesrecent results of Staden and Treglown.

Embedding spanning subgraphs in uniformly dense and inseparable graphs