#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

The edges of a graph have natural cyclic orderings. We investigate the matroids for which a similar cyclic ordering of the circuits is possible. A full characterization of the non-binary matroids with this property is given. Evidence of the difficulty of this problem for binary matroids is presented, along with a partial result for binary orderable matroids.

For a graph *G*, the ratio of |*E*(*G*)| to the minimum degree of *G* has a natural lower bound. For a matroid *M *that is representable over a finite field, we generalize this to a lower bound on the ratio of |*E*(*M*)| to the size of a smallest cocircuit of *M*. Further, we characterize the matroids that achieve equality in this bound.

Jamison and Mulder defined a graph *G* to be Θ_{3}-closed if, whenever vertices *x* and *y* of *G* are joined by three internally disjoint paths, *x* and *y* are adjacent. They found that graphs with this property can be built from cycles and complete graphs. We generalize this result to binary matroids, showing that the Θ_{3}-closed binary matroids can be built in a similar fashion from circuits, cycle matroids of complete graphs, and projective geometries.

#### Date

4-4-2023

#### Recommended Citation

Crenshaw, Cameron, "Matroid Generalizations of Some Graph Results" (2023). *LSU Doctoral Dissertations*. 6097.

https://digitalcommons.lsu.edu/gradschool_dissertations/6097

#### Committee Chair

Oxley, James G.