Extremal complete-graph graphic matroids

Establish that, for every n\geq 1, the graphic matroid of the complete graph on n vertices is an extremal matroid.

Background

The paper introduces extremal matroids as those whose points are vertices of the polytope Ωr,n\Omega_{r,n}. It asks for meaningful classes of extremal matroids and conjectures that complete-graph graphic matroids provide one such class. Existing characterizations by valuative invariants do not directly yield the required vertex result.

References

For every $n\geq 1$, the graphic matroid $\mathsf{K}_n$ of the complete graph on $n$ vertices is an extremal matroid.

— The polytope of all matroids  (2502.20157 - Ferroni et al., 27 Feb 2025) in Section 7.1, “Extremal matroids and the pursuit of counterexamples in matroid theory”