2024/12/13 by Jagdeep Singh, Singh, Jagdeep · 1 citation
Computer Science · #05B35 #94B65 #Advanced Algebra and Logic #Combinatorics (math.CO) #FOS: Mathematics #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2412.10326
openalex publication_date 2024/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.