Notes on embedding trees in graphs with O(|T|)-sized covers
Abstract
This is a companion paper to the paper "Hyperstability in the Erdos-Sos Conjecture". In that paper the following rough structure theorem was proved for graphs G containing no copy of a bounded degree tree T: from any such G, one can delete o(|G||T|) edges in order to get a subgraph all of whose connected components have a cover of order 3|T|. This theorem creates an incentive for studying graphs whose connected components have covers of order O(|T|) - and this is what will be explored here. It turns out that such graphs are amenable to regularity approaches which have been successful in studying dense T-free graphs. In this paper we will follow such an approach from the paper "On the Erdos-Sos conjecture for trees with bounded degree" by Besomi, Pavez-Signe, and Stein and show how it can be adapted from dense graphs to graphs with a small cover.
Keywords
Cite
@article{arxiv.2409.15189,
title = {Notes on embedding trees in graphs with O(|T|)-sized covers},
author = {Alexey Pokrovskiy},
journal= {arXiv preprint arXiv:2409.15189},
year = {2024}
}