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The Maximum Matroid of a Graph
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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.
Forward citations
Cited by 2 Pith papers
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