Matroids with many small circuits and cocircuits
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.
Publication Source (Journal or Book title)
Advances in Applied Mathematics
Oxley, J., Pfeil, S., Semple, C., & Whittle, G. (2019). Matroids with many small circuits and cocircuits. Advances in Applied Mathematics, 105, 1-24. https://doi.org/10.1016/j.aam.2018.12.005