Matroids with many small circuits and cocircuits

Abstract

Tutte proved that a non-empty 3-connected matroid with every element in a 3-element circuit and a 3-element cocircuit is either a whirl or the cycle matroid of a wheel. This result led to the Splitter Theorem. More recently, Miller proved that a matroid of sufficient size with every pair of elements in a 4-element circuit and a 4-element cocircuit is a tipless spike. Here we investigate matroids having similar restrictions on their small circuits and cocircuits. In particular, we completely determine the 3-connected matroids with every pair of elements in a 4-element circuit and every element in a 3-element cocircuit, as well as the 4-connected matroids with every pair of elements in a 4-element circuit and every element in a 4-element cocircuit.