Notes on embedding trees in graphs with O(|T|)-sized covers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909323185094656 |
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| author | Pokrovskiy, Alexey |
| author_facet | Pokrovskiy, Alexey |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Notes on embedding trees in graphs with O(|T|)-sized covers Pokrovskiy, Alexey Combinatorics 05C35, 05C05, 05C75 G.2.2 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. |
| title | Notes on embedding trees in graphs with O(|T|)-sized covers |
| topic | Combinatorics 05C35, 05C05, 05C75 G.2.2 |
| url | https://arxiv.org/abs/2409.15189 |