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
Large Circuits in Binary Matroids of Large Cogirth, II
✍ Scribed by Winfried Hochstättler; Bill Jackson
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 223 KB
- Volume
- 74
- Category
- Article
- ISSN
- 0095-8956
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✦ Synopsis
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  [5, 6, 7, 8], then M has a circuit of size at least min[r(M )+1, 2d ]. We conjecture that the latter result holds for all d 3. 1998 Academic Press (b) Suppose M is connected, e # E(M ), and every cocircuit Y of M with e  Y has size at least d. If M  P(F 7 , e) then M has a circuit C containing e and of size at least d+1.
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