The $k$-fold circuit property for matroids
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912899604152320 |
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| author | Jackson, Bill Nixon, Anthony Smith, Ben |
| author_facet | Jackson, Bill Nixon, Anthony Smith, Ben |
| contents | Double circuits were introduced by Lovász in 1980 as a fundamental tool in his derivation of a min-max formula for the size of a maximum matching in linear matroids. This formula was extended to all matroids satisfying the so-called `double circuit property' by Dress and Lovász in 1987. We extend these notions to $k$-fold circuits for all natural numbers $k$ and show, in particular that several families of matroids which are known to satisfy the double circuit property, satisfy the $k$-fold circuit property for all natural numbers $k$. These families include all pseudomodular matroids (such as full linear, algebraic and transversal matroids) and certain families of count matroids. These results suggest that the $k$-fold circuit property can be used as a measure of how close the lattice of flats of a matroid is to being a modular lattice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14782 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $k$-fold circuit property for matroids Jackson, Bill Nixon, Anthony Smith, Ben Combinatorics 05B35, 06C10, 90C27 Double circuits were introduced by Lovász in 1980 as a fundamental tool in his derivation of a min-max formula for the size of a maximum matching in linear matroids. This formula was extended to all matroids satisfying the so-called `double circuit property' by Dress and Lovász in 1987. We extend these notions to $k$-fold circuits for all natural numbers $k$ and show, in particular that several families of matroids which are known to satisfy the double circuit property, satisfy the $k$-fold circuit property for all natural numbers $k$. These families include all pseudomodular matroids (such as full linear, algebraic and transversal matroids) and certain families of count matroids. These results suggest that the $k$-fold circuit property can be used as a measure of how close the lattice of flats of a matroid is to being a modular lattice. |
| title | The $k$-fold circuit property for matroids |
| topic | Combinatorics 05B35, 06C10, 90C27 |
| url | https://arxiv.org/abs/2412.14782 |