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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On nonbinary $ 3$-connected matroids


Author: James G. Oxley
Journal: Trans. Amer. Math. Soc. 300 (1987), 663-679
MSC: Primary 05B35
MathSciNet review: 876471
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Abstract: It is well known that a matroid is binary if and only if it has no minor isomorphic to $ {U_{2,4}}$, the $ 4$-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid is in a $ {U_{2,4}}$-minor. The result was further extended by Seymour who showed that every pair of elements in a nonbinary $ 3$-connected matroid is in a $ {U_{2,4}}$-minor. This paper extends Seymour's theorem by proving that if $ \left\{ {x,\,y,\,z} \right\}$ is contained in a nonbinary $ 3$-connected matroid $ M$, then either $ M$ has a $ {U_{2,4}}$-minor using $ \left\{ {x,\,y,\,z} \right\}$, or $ M$ has a minor isomorphic to the rank-$ 3$ whirl that uses $ \left\{ {x,\,y,\,z} \right\}$ as its rim or its spokes.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0876471-1
PII: S 0002-9947(1987)0876471-1
Article copyright: © Copyright 1987 American Mathematical Society