Hochstättler, Winfried and Jackson, Bill (1998) Large Circuits in Binary Matroids of Large Cogirth: I.
Published in: Journal of Combinatorial Theory B Vol. 74 (1). pp. 35-52.

Abstract

Let F7 denote the Fano matroid and e be a fixed element of F7. Let P(F7,e) be the family of matroids obtained by taking the parallel connection of one or more copies of F7 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 not have an F7-minor, M is not F * 7 and d >= (r(M)+1)/2 then M has a circuit of size r(M)+1. We also show that if M is connected, e in E(M), M does not have both an F7-minor and an F * 7-minor, and M is not in P(F7,e), then M has a circuit containing e and of size at least d+1.

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