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Unavoidable Minors of Large 3-Connected Binary Matroids

✍ Scribed by Guoli Ding; Bogdan Oporowski; James Oxley; Dirk Vertigan

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
759 KB
Volume
66
Category
Article
ISSN
0095-8956

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✦ Synopsis

We show that, for every integer n greater than two, there is a number N such that every 3-connected binary matroid with at least N elements has a minor that is isomorphic to the cycle matroid of K 3, n , its dual, the cycle matroid of the wheel with n spokes, or the vector matroid of the binary matrix (I n | J n &I n ), where J n is the n_n matrix of all ones.

1996 Academic Press, Inc.

(1.3) Theorem. Let n be an integer greater than one. If M is a connected matroid with more than 4 n elements, then M contains a circuit or cocircuit with more than n elements.

πŸ“œ SIMILAR VOLUMES

Unavoidable Minors of Large 3-Connected
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Let F 7 denote the Fano matroid and e be a fixed element of F 7 . Let P(F 7 , e) be the family of matroids obtained by taking the parallel connection of one or more copies of F 7 about e. Let M be a simple binary matroid such that every cocircuit of M has size at least d 3. We show that if M does no

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Let F 7 denote the Fano matroid and M be a simple connected binary matroid such that every cocircuit of M has size at least d 3. We show that if M does not have an F 7 -minor, M{F\* 7 , and d Γ‚ [5, 6, 7, 8], then M has a circuit of size at least min[r(M )+1, 2d ]. We conjecture that the latter resul

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In this paper, we shall consider the following problem: up to duality, is a connected matroid reconstructible from its connectivity function? Cunningham conjectured that this question has an affirmative answer, but Seymour gave a counter-example for it. In the same paper, Seymour proved that a conne