Dillon Mayhew

Matroids with nine elements

We describe the computation of a catalogue containing all matroids with up to nine elements, and present some fundamental data arising from this catalogue. Our computation confirms and extends the results obtained in the 1960s by Blackburn, Crapo and Higgs. The matroids and associated data are stored in an on-line database, and we give three short examples of the use of this database.

On the asymptotic proportion of connected matroids

Very little is known about the asymptotic behavior of classes of matroids. We make a number of conjectures about such behaviors. For example, we conjecture that asymptotically almost every matroid: has a trivial automorphism group; is arbitrarily highly connected; and is not representable over any field. We prove one result: the proportion of labeled n-element matroids that are connected is asymptotically at least 1/2.

Wei-type duality theorems for matroids

We present several fundamental duality theorems for matroids and more general combinatorial structures. As a special case, these results show that the maximal cardinalities of fixed-ranked sets of a matroid determine the corresponding maximal cardinalities of the dual matroid. Our main results are applied to perfect matroid designs, graphs, transversals, and linear codes over division rings, in each case yielding a duality theorem for the respective class of objects.

A chain theorem for internally 4-connected binary matroids

Let M be a matroid. When M is 3-connected, Tuttes Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M)-E(N)|=1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M)-E(N)|=<3 unless M or its dual is the cycle matroid of a planar or Mobius quartic ladder, or a 16-element variant of such a planar ladder.

Matroid Complexity and Nonsuccinct Descriptions

We investigate an approach to matroid complexity that involves describing a matroid via a list of independent sets, bases, circuits, or some other family of subsets of the ground set. The computational complexity of algorithmic problems under this scheme appears to be highly dependent on the choice of input type. We define an order on the various methods of description, and we show how this order acts upon 10 types of input. We also show that under this approach several natural algorithmic problems are complete in classes thought not to be equal to P.

On excluded minors for real-representability

We show that for any infinite field K and any K-representable matroid N there is an excluded minor for K-representability that has N as a minor.

Stability, fragility, and Rota's Conjecture

Fix a matroid N. A matroid M is N-fragile if, for each element e of M, at least one of M@?e and M/e has no N-minor. The Bounded Canopy Conjecture is that all GF(q)-representable matroids M that have an N-minor and are N-fragile have branch width bounded by a constant depending only on q and N. A matroid N stabilizes a class of matroids over a field F if, for every matroid M in the class with an N-minor, every F-representation of N extends to at most one F-representation of M. We prove that, if Rota@?s Conjecture is false for GF(q), then either the Bounded Canopy Conjecture is false for GF(q) or there is an infinite chain of GF(q)-representable matroids, each not stabilized by the previous, each of which can be extended to an excluded minor. Our result implies the previously known result that Rota@?s Conjecture holds for GF(4), and that the classes of near-regular and sixth-roots-of-unity matroids have a finite number of excluded minors. However, the bound that we obtain on the size of such excluded minors is considerably larger than that obtained in previous proofs. For GF(5) we show that Rota@?s Conjecture reduces to the Bounded Canopy Conjecture.

An Obstacle to a Decomposition Theorem for Near-Regular Matroids

Seymours decomposition theorem [J. Combin. Theory Ser. B, 28 (1980), pp. 305-359] for regular matroids states that any matroid representable over both

MAXIMUM SIZE BINARY MATROIDS WITH NO AG(3,2)-MINOR ARE GRAPHIC ∗

\mathrm{GF}(2)

Inequivalent representations of matroids having no U 3,6 -minor

and