Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2507.12973 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866918522368557056 |
|---|---|
| author | Albrechtsen, Sandra Hamann, Matthias |
| author_facet | Albrechtsen, Sandra Hamann, Matthias |
| contents | We prove a coarse version of Halin's Grid Theorem: Every one-ended, locally finite graph that contains the disjoint union of infinitely many rays as an asymptotic minor also contains the half-grid as an asymptotic minor. More generally, we show that the same holds for arbitrary (not necessarily one-ended or locally finite) graphs under additional, necessary assumptions on the minor-models of the infinite rays. This resolves a conjecture of Georgakopoulos and Papasoglu.
As an application, we show that every one-ended, quasi-transitive, locally finite graph contains the half-grid as an asymptotic minor and as a diverging minor. This in particular includes all locally finite Cayley graphs of one-ended finitely generated groups and solves a problem of Georgakopoulos and Papasoglu. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12973 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A coarse Halin Grid Theorem with applications to quasi-transitive, locally finite graphs Albrechtsen, Sandra Hamann, Matthias Combinatorics Group Theory 05C83, 20F69, 05C63, 51F30, 05C40 We prove a coarse version of Halin's Grid Theorem: Every one-ended, locally finite graph that contains the disjoint union of infinitely many rays as an asymptotic minor also contains the half-grid as an asymptotic minor. More generally, we show that the same holds for arbitrary (not necessarily one-ended or locally finite) graphs under additional, necessary assumptions on the minor-models of the infinite rays. This resolves a conjecture of Georgakopoulos and Papasoglu. As an application, we show that every one-ended, quasi-transitive, locally finite graph contains the half-grid as an asymptotic minor and as a diverging minor. This in particular includes all locally finite Cayley graphs of one-ended finitely generated groups and solves a problem of Georgakopoulos and Papasoglu. |
| title | A coarse Halin Grid Theorem with applications to quasi-transitive, locally finite graphs |
| topic | Combinatorics Group Theory 05C83, 20F69, 05C63, 51F30, 05C40 |
| url | https://arxiv.org/abs/2507.12973 |