$k$-loose elements and $k$-paving matroids
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| Format: | Preprint |
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2024
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| _version_ | 1866915196937699328 |
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| author | Singh, Jagdeep |
| author_facet | Singh, Jagdeep |
| contents | For a matroid of rank $r$ and a non-negative integer $k$, an element is called $k$-loose if every circuit containing it has size greater than $r-k$. Zaslavsky and the author characterized all binary matroids with a $1$-loose element. In this paper, we establish a sharp linear bound on the size of a binary matroid, in terms of its rank, that contains a $k$-loose element. A matroid is called $k$-paving if all its elements are $k$-loose. Rajpal showed that for a prime power $q$, the rank of a $GF(q)$-matroid that is $k$-paving is bounded. We provide a bound on the rank of $GF(q)$-matroids that are cosimple and have two $k$-loose elements. Consequently, we deduce a bound on the rank of $GF(q)$-matroids that are $k$-paving. Additionally, we provide a bound on the size of binary matroids that are $k$-paving. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10326 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $k$-loose elements and $k$-paving matroids Singh, Jagdeep Combinatorics 05B35, 94B65 For a matroid of rank $r$ and a non-negative integer $k$, an element is called $k$-loose if every circuit containing it has size greater than $r-k$. Zaslavsky and the author characterized all binary matroids with a $1$-loose element. In this paper, we establish a sharp linear bound on the size of a binary matroid, in terms of its rank, that contains a $k$-loose element. A matroid is called $k$-paving if all its elements are $k$-loose. Rajpal showed that for a prime power $q$, the rank of a $GF(q)$-matroid that is $k$-paving is bounded. We provide a bound on the rank of $GF(q)$-matroids that are cosimple and have two $k$-loose elements. Consequently, we deduce a bound on the rank of $GF(q)$-matroids that are $k$-paving. Additionally, we provide a bound on the size of binary matroids that are $k$-paving. |
| title | $k$-loose elements and $k$-paving matroids |
| topic | Combinatorics 05B35, 94B65 |
| url | https://arxiv.org/abs/2412.10326 |