Bicircular Matroids with Circuits of at Most Two Sizes

Young in his paper titled, Matroid Designs in 1973, reports that Murty in his paper titled, Equicardinal Matroids and Finite Geometries in 1968, was the first to study matroids with all hyperplanes having the same size. Murty called such a matroid an ``Equicardinal Matroid''. Young renamed such a matroid a ``Matroid Design''. Further work on determining properties of these matroids was done by Edmonds, Murty, and Young in their papers published in 1972, 1973, and 1970 respectively. These authors were able to connect the problem of determining the matroid designs with specified parameters with results on balanced incomplete block designs. The dual of a matroid design is one in which all circuits have the same size. In 1971, Murty restricted his attention to binary matroids and was able to characterize all connected binary matroids having circuits of a single size. Lemos, Reid, and Wu in 2010, provided partial information on the class of connected binary matroids having circuits of two different sizes. They also shothat there are many such matroids. In general, there are not many results that specify the matroids with circuits of just a few different sizes. Cordovil, Junior, and Lemos provided such results on matroids with small circumference. Here we determine the connected bicircular matroids with all circuits having the same size. We also provide structural information on the connected bicircular matroids with circuits of two different sizes. The bicircular matroids considered are in general non-binary. Hence these results are a start on extending Murty's characterization of binary matroid designs to non-binary matroids.

Name thesis:degree_name

Ph.D. in Mathematics

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2010

Author dc:creator

• Lewis, Torina Deachune

Contributors dc:contributor

• Talmadge James Reid
• Haidong Wu
• Dawn Wilkins

• Bicircular
• Bicircular Matroid
• Dual Matroid Design
• Matroid Design
• Matroid Dissertation
• Single Circuit Spectrum
• Mathematics

Repository record dc:identifier

https://egrove.olemiss.edu/etd/1437

OAI identifier oai:identifier

oai:egrove.olemiss.edu:etd-2436