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.

10 pages