Bounding the number of bases of a matroid
Document Type
Article
Publication Date
1-1-2022
Abstract
Thomassen proved in 2010 that the number of spanning trees of a graph with vertex degrees d1,d2,…,dn is at most d1d2…dn−1. This note generalizes this result to show that if A is a matrix representing a rank-r matroid M over a field and S1,S2,…,Sr are the supports of the rows of A, then the number of bases of M is at most |S1||S2|…|Sr|. More generally, it is shown that if C1⁎,C2⁎,…,Cr⁎ are cocircuits of a rank-r matroid N such that the deletion of any k of these cocircuits from N drops the rank by at least k, then the number of bases of N is at most |C1⁎||C2⁎|…|Cr⁎|.
Publication Source (Journal or Book title)
Discrete Mathematics
Recommended Citation
Douthitt, J., & Oxley, J. (2022). Bounding the number of bases of a matroid. Discrete Mathematics, 345 (1) https://doi.org/10.1016/j.disc.2021.112636