Embedding spanning subgraphs in uniformly dense and inseparable graphs

We consider sufficient conditions for the existence of kth powers of Hamiltonian cycles in n-vertex graphs G with minimum degree (Formula presented.) for arbitrarily small (Formula presented.). About 20 years ago Komlós, Sarközy, and Szemerédi resolved the conjectures of Pósa and Seymour and obtained optimal minimum degree conditions for this problem by showing that (Formula presented.) suffices for large n. For smaller values of (Formula presented.) the given graph G must satisfy additional assumptions. We show that inducing subgraphs of density d > 0 on linear subsets of vertices and being inseparable, in the sense that every cut has density at least (Formula presented.), are sufficient assumptions for this problem and, in fact, for a variant of the bandwidth theorem. This generalizes recent results of Staden and Treglown.