Loose elements in binary and ternary matroids
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917892205838336 |
|---|---|
| author | Singh, Jagdeep Zaslavsky, Thomas |
| author_facet | Singh, Jagdeep Zaslavsky, Thomas |
| contents | We call a matroid element "loose" if it is contained in no circuits of size less than the rank of the matroid. A matroid in which all elements are loose is a paving matroid. Acketa determined all binary paving matroids, while Oxley specified all ternary paving matroids. We characterize the binary matroids that contain a loose element. For ternary matroids with a loose element, we show that their size is linear in terms of their rank. Moreover, for a prime power $q$, we give a partial characterization of $GF(q)$-representable matroids that have two or more loose elements; we note Rajpal's partial characterization of $GF(q)$-representable paving matroids as a consequence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07739 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Loose elements in binary and ternary matroids Singh, Jagdeep Zaslavsky, Thomas Combinatorics 05B35 We call a matroid element "loose" if it is contained in no circuits of size less than the rank of the matroid. A matroid in which all elements are loose is a paving matroid. Acketa determined all binary paving matroids, while Oxley specified all ternary paving matroids. We characterize the binary matroids that contain a loose element. For ternary matroids with a loose element, we show that their size is linear in terms of their rank. Moreover, for a prime power $q$, we give a partial characterization of $GF(q)$-representable matroids that have two or more loose elements; we note Rajpal's partial characterization of $GF(q)$-representable paving matroids as a consequence. |
| title | Loose elements in binary and ternary matroids |
| topic | Combinatorics 05B35 |
| url | https://arxiv.org/abs/2501.07739 |