Minors of matroids represented by sparse random matrices over finite fields
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866910301934321664 |
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| author | Gao, Pu Nelson, Peter |
| author_facet | Gao, Pu Nelson, Peter |
| contents | Consider a random $n\times m$ matrix $A$ over the finite field of order $q$ where every column has precisely $k$ nonzero elements, and let $M[A]$ be the matroid represented by $A$. In the case that q=2, Cooper, Frieze and Pegden (RS\&A 2019) proved that given a fixed binary matroid $N$, if $k\ge k_N$ and $m/n\ge d_N$ where $k_N$ and $d_N$ are sufficiently large constants depending on N, then a.a.s. $M[A]$ contains $N$ as a minor. We improve their result by determining the sharp threshold (of $m/n$) for the appearance of a fixed matroid $N$ as a minor of $M[A]$, for every $k\ge 3$, and every finite field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_15685 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Minors of matroids represented by sparse random matrices over finite fields Gao, Pu Nelson, Peter Combinatorics Consider a random $n\times m$ matrix $A$ over the finite field of order $q$ where every column has precisely $k$ nonzero elements, and let $M[A]$ be the matroid represented by $A$. In the case that q=2, Cooper, Frieze and Pegden (RS\&A 2019) proved that given a fixed binary matroid $N$, if $k\ge k_N$ and $m/n\ge d_N$ where $k_N$ and $d_N$ are sufficiently large constants depending on N, then a.a.s. $M[A]$ contains $N$ as a minor. We improve their result by determining the sharp threshold (of $m/n$) for the appearance of a fixed matroid $N$ as a minor of $M[A]$, for every $k\ge 3$, and every finite field. |
| title | Minors of matroids represented by sparse random matrices over finite fields |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2307.15685 |