Doctor of Philosophy (PhD)
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 Θ3-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 Θ3-closed binary matroids can be built in a similar fashion from circuits, cycle matroids of complete graphs, and projective geometries.
Crenshaw, Cameron, "Matroid Generalizations of Some Graph Results" (2023). LSU Doctoral Dissertations. 6097.
Oxley, James G.