A rank bound for bases and circuits in binary matroids
Abstract
Let b(M), d(M), and r(M) denote the number of bases, the number of circuits, and the rank of a matroid M, respectively. We prove that every nonempty simple binary matroid with no coloops satisfies 2b(M)≥(r(M)+1)d(M), with equality if and only if M is isomorphic to the Fano matroid. This confirms a conjecture recorded by Oxley in 1983. For the same class, we prove that deleting any element leaves at least as many bases as there are circuits in the original matroid: b(Mbackslash e)≥ d(M) for every e∈ E(M). We determine all equality cases and deduce a sharp linear lower bound for basis growth under successive series extensions. We also give a circuit-size exclusion condition guaranteeing distinct bases containing C−e for circuit-element pairs (C,e) with prescribed circuit sizes.
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BibTeX
@misc{fu2026rank,
title = {{A rank bound for bases and circuits in binary matroids}},
author = {Houshan Fu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11829},
eprint = {2610.11829},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11829},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A rank bound for bases and circuits in binary matroids
AU - Houshan Fu
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.11829
UR - https://arxiv.org/abs/2610.11829
AB - Let b(M), d(M), and r(M) denote the number of bases, the number of circuits, and the rank of a matroid M, respectively. We prove that every nonempty simple binary matroid with no coloops satisfies 2b(M)≥(r(M)+1)d(M), with equality if and only if M is isomorphic to the Fano matroid. This confirms a conjecture recorded by Oxley in 1983. For the same class, we prove that deleting any element leaves at least as many bases as there are circuits in the original matroid: b(Mbackslash e)≥ d(M) for every e∈ E(M). We determine all equality cases and deduce a sharp linear lower bound for basis growth under successive series extensions. We also give a circuit-size exclusion condition guaranteeing distinct bases containing C−e for circuit-element pairs (C,e) with prescribed circuit sizes.
ER -