Abstract
Analogous to the concept of uniquely pancyclic graphs, we define a uniquely pancyclic (UPC) matroid of rank r to be a (simple) rank-r matroid containing exactly one circuit of each length ℓ for 3≤ℓ≤r+1. Our discussion addresses the existence of graphic, binary, and transversal representations of UPC matroids. Using Shi’s results, which catalogued exactly seven non-isomorphic UPC graphs, we produce a nongraphic binary UPC matroid of rank 24. We consider properties of binary UPC matroids in general, and prove that all binary UPC matroids have a connectivity of 2.
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