On $k$-neighborly reorientations of oriented matroids
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916113632198656 |
|---|---|
| author | Hernández-Ortiz, Rangel Knauer, Kolja Montejano, Luis Pedro |
| author_facet | Hernández-Ortiz, Rangel Knauer, Kolja Montejano, Luis Pedro |
| contents | We study the existence and the number of $k$-neighborly reorientations of an oriented matroid. This leads to $k$-variants of McMullen's problem and Roudneff's conjecture, the case $k=1$ being the original statements on complete cells in arrangements. Adding to results of Larman and García-Colín, we provide new bounds on the $k$-McMullen's problem and prove the conjecture for several ranks and $k$ by computer. Further, we show that $k$-Roudneff's conjecture for fixed rank and $k$ reduces to a finite case analyse. As a consequence we prove the conjecture for odd rank $r$ and $k=\frac{r-1}{2}$ as well as for rank $6$ and $k=2$ with the aid of the computer. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00414 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On $k$-neighborly reorientations of oriented matroids Hernández-Ortiz, Rangel Knauer, Kolja Montejano, Luis Pedro Combinatorics We study the existence and the number of $k$-neighborly reorientations of an oriented matroid. This leads to $k$-variants of McMullen's problem and Roudneff's conjecture, the case $k=1$ being the original statements on complete cells in arrangements. Adding to results of Larman and García-Colín, we provide new bounds on the $k$-McMullen's problem and prove the conjecture for several ranks and $k$ by computer. Further, we show that $k$-Roudneff's conjecture for fixed rank and $k$ reduces to a finite case analyse. As a consequence we prove the conjecture for odd rank $r$ and $k=\frac{r-1}{2}$ as well as for rank $6$ and $k=2$ with the aid of the computer. |
| title | On $k$-neighborly reorientations of oriented matroids |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2306.00414 |