Elastic elements in 3-connected matroids

George Drummond, Zachary Gershkoff, Susan Jowett, Charles Semple, Jagdeep Singh

Published 2020-10-05Version 1

It follows by Bixby's Lemma that if $e$ is an element of a $3$-connected matroid $M$, then either $\mathrm{co}(M\backslash e)$, the cosimplification of $M\backslash e$, or $\mathrm{si}(M/e)$, the simplification of $M/e$, is $3$-connected. A natural question to ask is whether $M$ has an element $e$ such that both $\mathrm{co}(M\backslash e)$ and $\mathrm{si}(M/e)$ are $3$-connected. Calling such an element "elastic", in this paper we show that if $|E(M)|\ge 4$, then $M$ has at least four elastic elements provided $M$ has no $4$-element fans.