Notes on embedding trees in graphs with O(|T|)-sized covers

TL;DR

This paper explores how graphs with small covers of connected components can be analyzed using regularity methods, extending previous work on trees in graphs and their structural properties.

Contribution

It adapts regularity approach techniques from dense graphs to graphs with O(|T|)-sized covers of components, advancing understanding of tree embeddings in such graphs.

Findings

Graphs with small covers are amenable to regularity methods.

Extension of dense graph techniques to graphs with bounded cover sizes.

Potential for improved embedding results for trees in these graphs.

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.