Generalized Borsuk Graphs
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2021
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913542155796480 |
|---|---|
| author | Martinez-Figueroa, Francisco |
| author_facet | Martinez-Figueroa, Francisco |
| contents | Given a finite group $G$ acting freely on a compact metric space $M$, and $ε>0$, we define the $G$-Borsuk graph on $M$ by drawing edges $x\sim y$ whenever there is a non-identity $g\in G$ such that $d(x,gy)\leqε$. We show that when $ε$ is small, its chromatic number is determined by the topology of $M$ via its $G$-covering number, which is the minimum $k$ such that there is a closed cover $M=F_1\cup\dots\cup F_k$ with $F_i\cap g(F_i)=\emptyset$ for all $g\in G\setminus\{1\}$. We are interested in bounding this number. We give lower bounds using $G$-actions on Hom-complexes, and upper bounds using a recursive formula on the dimension of $M$. We conjecture that the true chromatic number coincides with the lower bound, and give computational evidence. We also study random $G$-Borsuk graphs, which are random induced subgraphs. For these, we compute thresholds for $ε$ that guarantee that the chromatic number is still that of the whole $G$-Borsuk graph. Our results are tight (up to a constant) when the $G$-index and dimension of $M$ coincide. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_06453 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Generalized Borsuk Graphs Martinez-Figueroa, Francisco Combinatorics Algebraic Topology Metric Geometry Probability Given a finite group $G$ acting freely on a compact metric space $M$, and $ε>0$, we define the $G$-Borsuk graph on $M$ by drawing edges $x\sim y$ whenever there is a non-identity $g\in G$ such that $d(x,gy)\leqε$. We show that when $ε$ is small, its chromatic number is determined by the topology of $M$ via its $G$-covering number, which is the minimum $k$ such that there is a closed cover $M=F_1\cup\dots\cup F_k$ with $F_i\cap g(F_i)=\emptyset$ for all $g\in G\setminus\{1\}$. We are interested in bounding this number. We give lower bounds using $G$-actions on Hom-complexes, and upper bounds using a recursive formula on the dimension of $M$. We conjecture that the true chromatic number coincides with the lower bound, and give computational evidence. We also study random $G$-Borsuk graphs, which are random induced subgraphs. For these, we compute thresholds for $ε$ that guarantee that the chromatic number is still that of the whole $G$-Borsuk graph. Our results are tight (up to a constant) when the $G$-index and dimension of $M$ coincide. |
| title | Generalized Borsuk Graphs |
| topic | Combinatorics Algebraic Topology Metric Geometry Probability |
| url | https://arxiv.org/abs/2110.06453 |