Large Circuits in Binary Matroids of Large Cogirth, I

✍ Scribed by Winfried Hochstättler; Bill Jackson

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 291 KB
• Volume: 74
• Category: Article
• ISSN: 0095-8956
• DOI: 10.1006/jctb.1998.1833

✦ Synopsis

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 not have an F 7 -minor, M{F 7 *, and d (r(M)+1)Â2 then M has a circuit of size r(M)+1. We also show that if M is connected, e # E(M), M does not have both an F 7 -minor and an F 7 *-minor, and M Â P(F 7 , e), then M has a circuit that contains e and has size at least d+1.