Matroids Representable Over Fields With a Common Subfield

Matroids Representable Over Fields With a Common Subfield

A matroid is GF(q)-regular if it is representable over all proper superfields of the field GF(q). We show that, for highly connected matroids having a large projective geometry over GF(q) as a minor, the property of GF(q)-regularity is equivalent to representability over both GF(q) and GF(q) for some odd integer t ≥ 3. We do this by means of an exact structural description of

Unlabeled equivalence for matroids representable over finite fields

Two r × n matrices A and A′ representing the same matroid M over GF (q), where q is a prime power, are projective equivalent representations of M if one can be obtained from the other by elementary row operations and column scaling. Bounds for projective inequivalence are difficult to obtain and are known for only a few special classes of matroids. In this paper we define two matrices A and A′ ...

On matroids representable over GF(3) and other fields

On Matroids Representable over Gf (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 ...

On pancyclic representable matroids

Bondy proved that an n-vertex simple Hamiltonian graph with at least n2/4 edges has cycles of every length unless it is isomorphic to Kn/2,n/2. This paper considers finding circuits of every size in GF (q)-representable matroids with large numbers of elements. A consequence of the main result is that a rank-r simple binary matroid with at least 2r−1 elements has circuits of all sizes or is isom...