  We consider sufficient conditions for the existence of $k$-th powers of
Hamiltonian cycles in $n$-vertex graphs $G$ with minimum degree $\mu n$ for
arbitrarily small $\mu>0$. About 20 years ago Koml\'os, Sark\"ozy, and
Szemer\'edi resolved the conjectures of P\'osa and Seymour and obtained optimal
minimum 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 $G$ 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 $\mu>0$, are sufficient assumptions for this
problem and, in fact, for a variant of the bandwidth theorem. This generalises
recent results of Staden and Treglown.
