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arXiv · 2607.09972

Free products, extremal matroids, and a generalization of perfect matroid designs

Joseph E. Bonin
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Abstract

Extremal matroids are those that yield vertices in the polytope of matroids. We show how to get the point, in the appropriate polytope of matroids, for the free product $M\square M'$ of matroids $M$ and $M'$ from the points for $M$ and $M'$ in their polytopes of matroids. With this we show that if $M\square M'$ is extremal, then $M$ and $M'$ are extremal. The converse is false. We identify a large class of matroids that includes perfect matroid designs and sparse paving matroids, and we find sufficient conditions under which free products of extremal matroids in this class are extremal.

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Joseph E. Bonin. 2026-07-10. Free products, extremal matroids, and a generalization of perfect matroid designs. https://arxiv.org/abs/2607.09972

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