This article studies the girth and cogirth problems for connected matroids. The problem of finding the cogirth of a graphic matroid has been intensively studied, but studies on the equivalent problem for a linear matroid or a general matroid have been rarely reported. Based on the duality and connectivity of a matroid, we prove properties associated with the girth and cogirth of a matroid whose...

Let ( ) M  denote the maximum number of disjoint bases in a matroid M . For a connected graph G , let ( ) = ( ( )) G M G   , where ( ) M G is the cycle matroid of G . The well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs G with ( ) G k   . Edmonds generalizes this theorem to matroids. In [1] and [2], for a matroid M with ( ) M k   , elements ( ) e E...

Matroids are combinatorial structures that generalize the notion of linear independence in matrices. There are many equivalent definitions of matroids, we will use one that focus on its independent sets. A matroid M is defined on a finite ground set E (or E(M) if we want to emphasize the matroid M) and a collection of subsets of E are said to be independent. The family of independent sets is de...

Matroids are combinatorial structures that generalize the notion of linear independence in matrices. There are many equivalent definitions of matroids, we will use one that focus on its independent sets. A matroid M is defined on a finite ground set E (or E(M) if we want to emphasize the matroid M) and a collection of subsets of E are said to be independent. The family of independent sets is de...

We study the positive Bergman complex B(M) of an oriented matroid M , which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid M . The positive Bergman complex is defined so that given a linear ideal I with associated oriented matroid MI , the positive tropical variety associated to I is equal to the fan over B(MI). Our main result is that a certain “fine” ...

A sufficiently large connected matroid M contains a big circuit or a big cocircuit. Pou-Lin Wu showed that we can ensure that M has a big circuit or a big cocircuit containing any chosen element of M . In this paper, we begin with a fixed connected matroid N and we take M to be a connected matroid that has N as a minor. Our main result establishes that ifM is sufficiently large, then, up to dua...

We prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M∗(K3,k), M(Wk), M(Kk), the cycle matroid of the graph obtained from K2,k by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K3,k by adding a complete graph on the vertex class with three vertices.

A link between matroid theory and p-branes is discussed. The Schild type action for p-branes and matroid bundle notion provide the two central structures for such a link. We use such a connection to bring the duality concept in matroid theory to p-branes physics. Our analysis may be of particular interest in M-theory and in matroid bundle theory.

We prove that, for each nonnegative integer k and each matroid N , if M is a 3-connected matroid containing N as a minor, and the the branch width of M is sufficiently large, then there is a k-element set X ⊆ E(M) such that one of M \X and M/X is 3-connected and contains N as a minor.

A link between matroid theory and p-branes is discussed. The Schild type action for p-branes and matroid bundle notion provide the two central structures for such a link. We use such a connection to bring the duality concept in matroid theory to p-branes physics. Our analysis may be of particular interest in M-theory and in matroid bundle theory.