A characterization of the ternary matroids with no M(K 4)-minor

Abstract

Brylawski identified the class of binary matroids with no minor isomorphic to M( K 4) as being the class of series-parallel networks. From this he deduced that, for all such matroids M, the critical exponent c( M; 2) is at most 2. He also conjectured that a similar result is true over all finite fields GF( q). This paper examines the classes of ternary and GF(4)-representable matroids with no M(K 4 )-minor. The main result characterizes the former class by showing that, with one exception, the only non-trivial 3-connected members of this class are whirls or minors of the Steiner system S(5, 6, 12). This characterization is then used to show that, for all ternary matroids M with no M(K 4 )-minor, c( M; 3)≤2, thereby verifying Brylawski's conjecture in the case that q = 3. The characterization is also used to give excluded-minor descriptions for the class of ternary gammoids and two other related classes. The first of these results answers a question of Ingleton and verifies another conjecture of Brylawski.