Embedding spanning subgraphs in uniformly dense and inseparable graphs

Abstract

We consider sufficient conditions for the existence of k-th powers of Hamiltonian cycles in n-vertex graphs G with minimum degree µn for arbitrarily small µ ą 0.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 µ " k k`1 suffices for large n. For smaller values of µ the given graph G must satisfy additional assumptions. We show that inducing subgraphs of density d ą 0 on linear subsets of vertices and being inseparable, in the sense that every cut has density at least µ ą 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.