On density-critical matroids
For a matroid M having m rank-one flats, the density d(M) isr(M)m unless m = 0, in which case d(M) = 0. A matroid is density-critical if all of its proper minors of non-zero rank have lower density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among simple matroids that cannot be covered by k independent sets is density-critical. It is straightforward to show that U1,k+1 is the only minor-minimal loopless matroid with no covering by k independent sets. We prove that there are exactly ten minor-minimal simple obstructions to a matroid being able to be covered by two independent sets. These ten matroids are precisely the densitycritical matroids M such that d(M) > 2 but d(N) ≤ 2 for all proper minors N of M. All density-critical matroids of density less than 2 are series-parallel networks. For k ≥ 2, although ﬁnding all density-critical matroids of density at most k does not seem straightforward, we do solve this problem for k = 9 4.
Publication Source (Journal or Book title)
Electronic Journal of Combinatorics
Campbell, R., Grace, K., Oxley, J., & Whittle, G. (2020). On density-critical matroids. Electronic Journal of Combinatorics, 27 (2), 1-16. https://doi.org/10.37236/8584