Maximal matroids and counterexamples

Abstract

Jackson and Tanigawa conjectured that the rigidity matroid ${\mathcal{R}}_n^d$ and the hyperconnectivity matroid ${\mathcal{H}}_n^d$ are the unique maximal matroids in the posets of $\{K_{d+2},K_{d+2,d+2}\}$-matroids and $\{K_{d+2},K_{d+1,d+1}\}$-matroids, respectively. We disprove these conjectures by showing the existence of maximal matroids that are distinct from the proposed candidates.

arXiv comment10 pages. v2 adds more details on circuits being preserved in the Hyperconnectivity matroid with respect to the coning operation