Finite Matroid Research Articles

On generalisations of the Aharoni–Pouzet base exchange theorem

AbstractThe Greene–Magnanti theorem states that if is a finite matroid, and are bases and is a partition, then there is a partition such that is a base for every . The special case where each is a singleton can be rephrased as the existence of a perfect matching in the base transition graph. Pouzet conjectured that this remains true in infinite‐dimensional vector spaces. Later, he and Aharoni answered this conjecture affirmatively not just for vector spaces but also for infinite matroids. We prove two generalisations of their result. On the one hand, we show that ‘being a singleton’ can be relaxed to ‘being finite’ and this is sharp in the sense that the exclusion of infinite sets is really necessary. In addition, we prove that if and are bases, then there is a bijection between their finite subsets such that is a base for every . In contrast to the approach of Aharoni and Pouzet, our proofs are completely elementary, they do not rely on infinite matching theory.

A formula for simplicial tree-numbers of matroid complexes

We give a formula for the simplicial tree-numbers of the independent set complex of a finite matroid M as a product of eigenvalues of the total combinatorial Laplacians on this complex. Two matroid invariants emerge naturally in describing the multiplicities of these eigenvalues in the formula: one is the unsigned reduced Euler characteristic of the independent set complex and the other is the β-invariant of a matroid. We will demonstrate various applications of this formula including a “matroid theoretic” derivation of Kalai’s simplicial tree-numbers of a standard simplex.

Matroid intersection, base packing and base covering for infinite matroids

As part of the recent developments in infinite matroid theory, there have been a number of conjectures about how standard theorems of finite matroid theory might extend to the infinite setting. These include base packing, base covering, and matroid intersection and union. We show that several of these conjectures are equivalent, so that each gives a perspective on the same central problem of infinite matroid theory. For finite matroids, these equivalences give new and simpler proofs for the finite theorems corresponding to these conjectures. This new point of view also allows us to extend, and simplify the proofs of some cases where these conjectures were known to be true.

A generalization of the Burnside basis theorem

A B-group is a group such that all its minimal generating sets (with respect to inclusion) have the same size. We prove that the class of finite B-groups is closed under taking quotients and that every finite B-group is solvable. Via a complete classification of Frattini-free finite B-groups we obtain a general structure theorem for finite B-groups. Applications include new proofs for the characterization of finite matroid groups.

Finite connectivity in infinite matroids

We introduce a connectivity function for infinite matroids with properties similar to the connectivity function of a finite matroid, such as submodularity and invariance under duality. As an application we use it to extend Tutte’s Linking Theorem to finitary and cofinitary matroids.

Some Properties of the Closure Operator of a Pi-space

In this paper, we generalize the definition of a closure operator for a finite matroid to a pi-space and obtain the corresponding closure axioms. Then we discuss some properties of pi-spaces using the closure axioms and prove the non-existence for the dual of a pi-space. We also present some results on the automorphism group of a pi-space.

On the basis graph of a matroid of arbitrary cardinality

This paper first generalizes the notion of basis graph of a finite matroid to that of a matroid of arbitrary cardinality. Then we follow by dealing with the relationship of a matroid of arbitrary cardinality between its connectivity and its basis graph.

Expected value of the minimal basis of random matroid and distributions of q-analogs of order statistics

Abstract We formulate and prove a formula to compute the expected value of the minimal random basis of an arbitrary finite matroid whose elements are assigned weights which are independent and uniformly distributed on the interval [0, 1]. This metod yields an exact formula in terms of the Tutte polynomial. We give a simple formula to find the minimal random basis of the projective geometry P G ( r − 1 , q ) . In the second part we give a distribution and asymptotic distributions of q-analogs of the k-th order statistics and the intermediate order statistics with r − k → ∞ when n is a number of elements of the projective geometry P G ( r − 1 , q ) . The proofs will appear in a forthcoming publications.

A Convolution Formula for the Tutte Polynomial

Let M be a finite matroid with rank function r. We will write A M when we mean that A is a subset of the ground set of M, and write M|A and M A for the matroids obtained by restricting M to A and contracting M on A respectively. Let M* denote the dual matroid to M. (See [1] for definitions). The main theorem is Theorem 1. The Tutte polynomial TM(x, y) satisfies TM(x, y)= : A M TM|A(0, y) TM A(x, 0). (1)

A short proof of a theorem of vámos on matroid representations

Publisher Summary Vamos proved that if a (finite or infinite) matroid is representable over fields of arbitrarily large characteristic then it is representable over a field of characteristic zero. The method of proof is algebraic. Although the proof of this important result is by no means long, it may be of interest to see how ultraproducts can be used to give a very short and direct proof of it. This is the aim in this chapter. It also gives a brief proof of another result because of Vamos—namely, that if each finite subset of a matroid is representable over an extension field of a given field then so is the matroid itself.