On Maximum-Sized Golden-Mean Matroids

<p>A rank-r simple matroid is maximum-sized in a class if it has the largest number of elements out of all simple rank-r matroids in that class. Maximum-sized matroids have been classified for various classes of matroids: regular (Heller, 1957); dyadic (Kung and Oxley, 1988-90); k-regular (Semple, 1998); near-regular and sixth-root-of-unity (Oxley, Vertigan, and Whittle, 1998). Golden-mean matroids are matroids that are representable over the golden-mean partial field. Equivalently, a golden-mean matroid is a matroid that is representable over GF(4) and GF(5). Archer conjectured that there are three families of maximum-sized golden-mean matroids. This means that a proof of Archer’s conjecture is likely to be significantly more complex than the proofs of existing maximum-sized characterisations, as they all have only one family. In this thesis, we consider the four following subclasses of golden-mean matroids: those that are lifts of regular matroids, those that are lifts of nearregular matroids, those that are golden-mean-graphic, and those that have a spanning clique. We close each of these classes under minors, and prove that Archer’s conjecture holds in each of them. It is anticipated that the last of our theorems will lead to a proof of Archer’s conjecture for golden-mean matroids of sufficiently high rank.</p>

On Maximum-Sized Golden-Mean Matroids

<p>A rank-r simple matroid is maximum-sized in a class if it has the largest number of elements out of all simple rank-r matroids in that class. Maximum-sized matroids have been classified for various classes of matroids: regular (Heller, 1957); dyadic (Kung and Oxley, 1988-90); k-regular (Semple, 1998); near-regular and sixth-root-of-unity (Oxley, Vertigan, and Whittle, 1998). Golden-mean matroids are matroids that are representable over the golden-mean partial field. Equivalently, a golden-mean matroid is a matroid that is representable over GF(4) and GF(5). Archer conjectured that there are three families of maximum-sized golden-mean matroids. This means that a proof of Archer’s conjecture is likely to be significantly more complex than the proofs of existing maximum-sized characterisations, as they all have only one family. In this thesis, we consider the four following subclasses of golden-mean matroids: those that are lifts of regular matroids, those that are lifts of nearregular matroids, those that are golden-mean-graphic, and those that have a spanning clique. We close each of these classes under minors, and prove that Archer’s conjecture holds in each of them. It is anticipated that the last of our theorems will lead to a proof of Archer’s conjecture for golden-mean matroids of sufficiently high rank.</p>

Critical Exponents, Colines, and Projective Geometries

In [9, p. 469], Oxley made the following conjecture, which is a geometric analogue of a conjecture of Lovász (see [1, p. 290]) about complete graphs.Conjecture 1.1.Let G be a rank-n GF(q)-representable simple matroid with critical exponent n − γ. If, for every coline X in G, c(G/X; q) = c(G; q) − 2 = n − γ − 2, then G is the projective geometry PG(n − 1, q).We shall call the rank n, the critical ‘co-exponent’ γ, and the order q of the field the parameters of Oxley's conjecture. We exhibit several counterexamples to this conjecture. These examples show that, for a given prime power q and a given positive integer γ, Oxley's conjecture holds for only finitely many ranks n. We shall assume familiarity with matroid theory and, in particular, the theory of critical problems. See [6] and [9].A subset C of points of PG(n − 1, q) is a (γ, k)-cordon if, for every k-codimensional subspace X in PG(n − 1, q), the intersection C ∩ X contains a γ-dimensional subspace of PG(n − 1, q). In this paper, our primary interest will be in constructing (γ, 2)-cordons. With straightforward modifications, our methods will also yield (γ, k)-cordons.Complements of counterexamples to Oxley's conjecture are (γ, 2)-cordons.

Sign-Coherent Identities for Characteristic Polynomials of Matroids

We derive an explicit formula for the difference χ(G;λ) − χ(G|X;λ)χ(Tx(G); λ)/(λ − 1), where χ(G;λ) is the characteristic polynomial of a simple matroid G, G|X is the restriction of G to a flat X in G, and Tx(G) is the complete principal truncation of G at the flat X. Two counting proofs of this formula are given. The first uses the critical problem and the second uses the broken-circuit complex. We also derive several inequalities involving Whitney numbers of the first kind and other numerical invariants.