We consider sufficient conditions for the existence of kth powers of Hamiltonian cycles in n-vertex graphs G with minimum degree n for arbitrarily small > 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 = k k+1 suffices for large n. For smaller values of the given graph G must satisfy additional assumptions. We show that inducing subgraphs of density > 0 on linear subsets of vertices and being inseparable, in the sense that every cut has density at least > 0, are sufficient assumptions for this problem and, in fact, for a variant of the bandwidth theorem. This generalizes recent results of Staden and Treglown. KEYWORDS absorption method, bandwidth theorem, powers of Hamiltonian cycles 1 INTRODUCTION We study sufficient conditions for the existence of spanning subgraphs in large finite graphs and begin the discussion with powers of Hamiltonian cycles. For k ∈ N the kth power of a given graph H is the graph H k on the same vertex set with xy being an edge in H k if x and y are distinct vertices of H that are connected in H by a path of at most k edges. For simplicity, we refer to a kth power of a path with at least k vertices as a k-path. Moreover, we refer to the ordered k-tuples of the first and last k vertices This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes.

The interplay of minimum degree conditions and structural properties of large graphs with forbidden subgraphs is a central topic in extremal graph theory. For a given graph F we define the homomorphism threshold as the infimum over all α P r0, 1s such that every n-vertex F -free graph G with minimum degree at least αn has a homomorphic image H of bounded order (i.e. independent of n), which is F -free as well. Without the restriction of H being F -free we recover the definition of the chromatic threshold, which was determined for every graph F by Allen et al. [Adv. Math. 235 (2013), 261-295]. The homomorphism threshold is less understood and we address the problem for odd cycles.

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.