Embedding spanning subgraphs in uniformly dense and inseparable graphs

TL;DR

This paper establishes new sufficient conditions involving density and inseparability for the existence of powers of Hamiltonian cycles in dense graphs, extending classical results and generalizing recent findings.

Contribution

It introduces conditions based on subgraph density and inseparability that ensure Hamiltonian cycle powers in graphs with small minimum degree, broadening previous theorems.

Findings

Inducing subgraphs of positive density on linear vertex sets suffice.

Inseparability with minimum cut density guarantees Hamiltonian powers.

Results extend and generalize recent theorems in graph embedding.

Abstract

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.