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(k\\)-loose elements and \\(k\\)-paving matroids
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(k\\)-loose elements and \\(k\\)-paving matroids
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Paper
(k\\)-loose elements and \\(k\\)-paving matroids
2025
Overview
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.
Publisher
Cornell University Library, arXiv.org
Subject
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