Jennifer McNulty

Connected hyperplanes in binary matroids

In this paper, we prove that any simple and cosimple connected binary matroid has at least four connected hyperplanes. We further prove that each element in such a matroid is contained in at least two connected hyperplanes. Our main result generalizes a matroid result of Kelmans, and independently, of Seymour. The following consequence of the main result generalizes a graph result of Thomassen and Toft on induced non-separating cycles and another graph result of Kaugars on deletable vertices. If G is a simple 2-connected graph with minimum degree at least 3, then, for every edge e, there are at least two induced non-separating cycles avoiding e and two deletable vertices non-incident to e. Moreover, G has at least four induced non-separating cycles.

The Matroid Ramsey Number n (6,6)

A tight upper bound on the number of elements in a connected matroid with fixed rank and largest cocircuit size is given. This upper bound is used to show that a connected matroid with at least thirteen elements contains either a circuit or a cocircuit with at least six elements. In the language of matroid Ramsey numbers, n(6, 6) = 13: this is the largest currently known matroid Ramsey number.

Note: On cocircuit covers of bicircular matroids

Given a graph G, one can define a matroid M=(E,C) on the edges E of G with circuits C where C is either the cycles of G or the bicycles of G. The former is called the cycle matroid of G and the latter the bicircular matroid of G. For each bicircular matroid B(G), we find a cocircuit cover of size at most the circumference of B(G) that contains every edge at least twice. This extends the result of Neumann-Lara, Rivera-Campo and Urrutia for graphic matroids.

Fixing Numbers for Matroids

Motivated by work in graph theory, we define the fixing number for a matroid. We give upper and lower bounds for fixing numbers for a general matroid in terms of the size and maximum orbit size (under the action of the matroid automorphism group). We prove the fixing numbers for the cycle matroid and bicircular matroid associated with 3-connected graphs are identical. Many of these results have interpretations through permutation groups, and we make this connection explicit.