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The Maximum Matroid of a Graph

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arxiv 1910.05390 v2 pith:J62T2YJO submitted 2019-10-11 math.CO

Meera Sitharam , Andrew Vince This is my paper · ORCID
classification math.CO
keywords matroidmaximumgraphcompletecycledeterminerelatedbases
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abstract

The ground set for all matroids in this paper is the set of all edges of a complete graph. The notion of a {\it maximum matroid for a graph} $G$ is introduced, and the existence and uniqueness of the maximum matroid for any graph $G$ is proved. The maximum matroid for $K_3$ is shown to be the cycle (or graphic) matroid. This result is pursued in two directions - to determine the maximum matroid for the $m$-cycle $C_m$ and to determine the maximum matroid for the complete graph $K_m$. The maximum matroid for $K_4$ is the matroid whose bases are the Laman graphs, related to structural rigidity of frameworks in the plane. The maximum matroid for $K_5$ is related to a famous 153 year old open problem of J. C. Maxwell.

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  1. Algebraic Geometry of Cactus, Pascal, and Pappus Matroids

    math.CO 2025-06 conditional novelty 6.0 of 10

    For cactus, Pascal, and Pappus configurations, the matroid ideal is generated, up to radical, by circuit, Grassmann-Cayley, and liftability polynomials.

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