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[PAST EVENT] Colloquium/CSUMS Talk
September 14, 2012
2pm - 3pm
Matroids provide an abstract representation of linear independence. They also provide a generalization of graphs with the benefit of a well defined dual. The matroid typically associated with a given graph is based on the graph's circuits. Here we consider an alternative, based on the graph's bicycles. This talk will require no previous knowledge of matroids and will be accessible to undergraduates.
Matroids with all hyperplanes having the same size, called "Equicardinal Matroids" or "Matroid Designs", are connected to combinatorial objects known as balanced incomplete block designs. The dual of a matroid design is a matroid in which all circuits have the same size. A few small classes of such matroids have been characterized. In this talk we present a recent characterization of all bicircular matroids with circuits of a single size, and all those with circuits of exactly two sizes. The techniques involved are graph theoretic.
Matroids with all hyperplanes having the same size, called "Equicardinal Matroids" or "Matroid Designs", are connected to combinatorial objects known as balanced incomplete block designs. The dual of a matroid design is a matroid in which all circuits have the same size. A few small classes of such matroids have been characterized. In this talk we present a recent characterization of all bicircular matroids with circuits of a single size, and all those with circuits of exactly two sizes. The techniques involved are graph theoretic.