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arXiv · 1909.13071

Embedding spanning subgraphs in uniformly dense and inseparable graphs

Oliver Ebsen·Giulia S. Maesaka·Christian Reiher·Mathias Schacht·Bjarne Schülke
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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.

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Oliver Ebsen, Giulia S. Maesaka, Christian Reiher, Mathias Schacht, Bjarne Schülke. 2019-09-28. Embedding spanning subgraphs in uniformly dense and inseparable graphs. https://arxiv.org/abs/1909.13071

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