A Note on the Critical Problem for Matroids

Let M be a matroid representable over GF(q) and S be a subset of its ground set. In this note we prove that S is maximal with the property that the critical exponent c(M|S; q) does not exceed k if and only if S is maximal with the property that c(M · S) ≤ k. In addition, we show that, for regular matroids, the corresponding result holds for the chromatic number. © 1984, Academic Press Inc. (London) Limited. All rights reserved

Excluding Kuratowski graphs and their duals from binary matroids

We consider some applications of our characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor. We characterise the internally 4-connected binary matroids with no minor in some subset of {M(K3,3),M*(K3,3),M(K5),M*(K5)} that contains either M(K3,3) or M*(K3,3). We also describe a practical algorithm for testing whether a binary matroid has a minor in the subset. In addition we characterise the growth-rate of binary matroids with no M(K3,3)-minor, and we show that a binary matroid with no M(K3,3)-minor has critical exponent over GF(2) at most equal to four.Comment: Some small change