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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.20255 |
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| _version_ | 1866914175585878016 |
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| author | Hernández, Rangel Montejano, Luis Pedro |
| author_facet | Hernández, Rangel Montejano, Luis Pedro |
| contents | In this paper we study the number of $k$-neighborly reorientations of an oriented matroid, leading to study $k$-Roudneff's conjecture, the case $k=1$ being the original statement conjectured in 1991. We first prove the conjecture for the family of Lawrence oriented matroids (LOMs) with even rank $r=2k+2$ and also for low ranks by computer. Next, we provide a general upper bound for the number of $k$-neighborly reorientations of any LOM. Finally, we prove that for any $k\ge 1$ and any oriented matroid on $n$ elements, $k$-Roudneff's conjecture holds asymptotically as $n\rightarrow \infty$ and thus giving more credit to the conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20255 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New results on $k$-Roudneff's conjecture Hernández, Rangel Montejano, Luis Pedro Combinatorics In this paper we study the number of $k$-neighborly reorientations of an oriented matroid, leading to study $k$-Roudneff's conjecture, the case $k=1$ being the original statement conjectured in 1991. We first prove the conjecture for the family of Lawrence oriented matroids (LOMs) with even rank $r=2k+2$ and also for low ranks by computer. Next, we provide a general upper bound for the number of $k$-neighborly reorientations of any LOM. Finally, we prove that for any $k\ge 1$ and any oriented matroid on $n$ elements, $k$-Roudneff's conjecture holds asymptotically as $n\rightarrow \infty$ and thus giving more credit to the conjecture. |
| title | New results on $k$-Roudneff's conjecture |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.20255 |