New excluded minors for the class $\mathcal{M}_3$ of regular matroids

Abstract

Engel, de Gaay Fortman, and Schreieder attach to each prime $\ell$ a minor-closed class $\mathcal{M}_\ell$ of regular matroids, whose excluded minors govern the failure of the integral Hodge conjecture for curve classes on very general principally polarized abelian varieties. For $\ell=2$ the class is the cographic matroids, with excluded minors $M(K_5)$ and $M(K_{3,3})$ by Tutte's theorem; for $\ell=3$ the only excluded minor explicitly identified so far is $M(K_{3,5})$, and the general characterization is [EGFS, Problem 8.8]. We exhibit five new excluded minors for $\mathcal{M}_3$, of ranks 8, 9, 9, 9, and 10; none contains $M(K_{3,5})$ or any of the other four as a minor, so the excluded-minor list of Problem 8.8 has at least six members, with excluded minors at every rank from 7 through 10. Combined with $M(K_7) \notin \mathcal{M}_3$, established by the same authors, and the minor-closedness of $\mathcal{M}_3$, some minor of $M(K_7)$ of rank between 4 and 6 is a further excluded minor, so the list has at least seven members, six of them explicitly identified. The list is structurally diverse: two bipartite apex constructions with automorphism group $S_4$, and three non-bipartite, non-planar rank-9 graphs with no such apex structure. Every verdict is certified by a finite $\mathbb{F}_3$-linear-algebra computation with explicit machine-checkable witnesses, cross-checked by independent implementations; the certificates further show that the five matroids lie outside the larger class $\widetilde{\mathcal{M}}_3$. We also record what are, to our knowledge, the first $\ell=5$ data on this corank-8 slice: $M(K_{3,5})$ and the rank-8 and rank-10 minors lie in $\mathcal{M}_5$ by explicit certificates, and the radical distance $d(M(K_{3,5}))=6$ is determined exactly.