Conjectures on Circuits, Clones, and Connectivity in Matroids

Document Type

Presentation Abstract

Presentation Date

11-17-2006

Abstract

Two elements of a matroid are clones if the map that interchanges the two elements and fixes all other elements is an automorphism. Clones are important in the study of matroid representability. We give results on the number and size of clone sets in representable matroids.

Smith conjectured in 1979 that two distinct cycles in a k-connected graph meet in at least k vertices when k >= 2. Matroid extensions of this conjecture are considered. We also give a result that characterizes the connected binary matroids with two different circuit sizes.

Finally, we consider conjectures of Wu on deletable and contractible elements in 4-connected matroids. This is the result of work with many others such as Cotwright, Lemos, McMurray, Robbins, Sheppardson, Wei, Wu, and Zhou.

Additional Details

Part of the 2006 Montana Matroid Workshop

Friday, 17 November 2006
2:10 p.m. in Math 211

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