Geodesic
Elastic elements in 3-connected matroids
George Drummond1 ; Zach Gershkoff2 ; Susan Jowett3 ; Charles Semple1 ; Jagdeep Singh2
1University of Canterbury
2Louisiana State University
3Victoria University of Wellington
The electronic journal of combinatorics, Tome 28 (2021) no. 2
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It follows by Bixby's Lemma that if $e$ is an element of a $3$-connected matroid $M$, then either $\textrm{co}(M\backslash e)$, the cosimplification of $M\backslash e$, or $\textrm{si}(M/e)$, the simplification of $M/e$, is $3$-connected. A natural question to ask is whether $M$ has an element $e$ such that both $\textrm{co}(M\backslash e)$ and $\textrm{si}(M/e)$ 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. A corrigendum was added to this paper on December 3, 2021.
DOI : 10.37236/9919
Classification : 05B35
Mots-clés : Bixby's lemma, \(k\)-connected matroid

George Drummond  1   ; Zach Gershkoff  2   ; Susan Jowett  3   ; Charles Semple  1   ; Jagdeep Singh  2

1 University of Canterbury
2 Louisiana State University
3 Victoria University of Wellington 
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George Drummond; Zach Gershkoff; Susan Jowett; Charles Semple; Jagdeep Singh. Elastic elements in 3-connected matroids. The electronic journal of combinatorics, Tome 28 (2021) no. 2. doi: 10.37236/9919

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