Date of Award
Ph.D. in Mathematics
Jason D. Hoeksema
Talmadge James Reid
Matroids designs are defined to be matroids in which the hyperplanes all have the same size. The dual of a matroid design is a matroid with all circuits of the same size, called a dual matroid design. The connected bicircular dual matroid designs have been characterized previously. In addition, these results have been extended to connected bicircular matroids with circuits of two sizes in the case that the associated graph is a subdivision of a 3-connected graph. In this dissertation, we will use a graph theoretic approach to discuss the characterizations of bicircular matroids with circuits of two and three sizes. We will characterize the associated graph of a bicircular matroid with circuits of two sizes. Moreover, we will provide a characterization of connected bicircular matroids with circuits of three sizes in the case that the associated graph is a subdivision of a 3-connected graph. We will also investigate the circuit spectrum of bicircular matroids whose associated graphs have minimum degree at least i for k ≥ 1, and show that there exists a set of bicycles with consecutive bicycle lengths.
Putnam, Bette Catherine, "The Characterization Of Graphs With Small Bicycle Spectrum" (2014). Electronic Theses and Dissertations. 681.