Geoff Whittle

Branch-Width and Well-Quasi-Ordering in Matroids and Graphs

We prove that a class of matroids representable over a fixed finite field and with bounded branch-width is well-quasi-ordered under taking minors. With some extra work, the result implies Robertson and Seymours result that graphs with bounded tree-width (or equivalently, bounded branch-width) are well-quasi-ordered under taking minors. We will not only derive their result from our result on matroids, but we will also use the main tools for a direct proof that graphs with bounded branch-width are well-quasi-ordered under taking minors. This proof also provides a model for the proof of the result on matroids, with all specific matroid technicalities stripped off.

On Matroids Representable over (3) and Other Fields

The matroids that are representable over GF (3) and some other fields depend on the choice of field. This paper gives matrix characterisations of the classes that arise. These characterisations are analogues of the characterisation of regular matroids as the ones that can be represented over the rationals by a totally-unimodular matrix. Some consequences of the theory are as follows. A matroid is representable over GF (3) and GF (5) if and only if it is representable over GF (3) and the rationals, and this holds if and only if it is representable over GF (p) for all odd primes p. A matroid is representable over GF (3) and the complex numbers if and only if it is representable over GF (3) and GF (7). A matroid is representable over GF (3), GF (4) and GF (5) if and only if it is representable over every field except possibly GF (2). If a matroid is representable over GF (p) for all odd primes p, then it is representable over

Addendum to matroid tree-width

Hliněný and Whittle have shown that the traditional tree-width notion of a graph can be defined without an explicit reference to vertices, and that it can be naturally extended to all matroids. Unfortunately their original paper [P. Hliněný, G. Whittle, Matroid tree-width, European J. Combin. 27 (2006) 1117-1128], as pointed out by Isolde Adler in 2007, contained some incorrect arguments. It is the purpose of this addendum to correct the affected proofs. (All the theorems and results of the original paper remain valid.)

D-Dimensional hypercubes and the Euler and MacNeish conjectures

We give a construction for large sets of mutually orthogonal hypercubes of dimensionald given sets of mutually orthogonal latin squares and hypercubes of lower dimension. We also considerd>-2 dimensional versions of the Euler and MacNeish conjectures as well as discussing applications to improved constructions of (t, m, s)-nets, useful in pseudorandom number generation and quasi-Monte-Carlo methods of numerical integration.

Inequivalent representations of ternary matroids

Abstract This paper considers representations of ternary matroids over fields other than GF(3). It is shown that a 3-connected ternary matroid representable over a finite field F has at most ¦F¦ - 2 inequivalent representations over F. This resolves a special case of a conjecture of Kahn in the affirmative.