On size, circumference and circuit removal in 3-connected matroids |
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Authors: | Manoel Lemos James Oxley |
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Affiliation: | a Departamento de Matemática, Universidade Federal de Pernambuco, Recife, Pernambuco 50740-540, Brazil b Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918, USA |
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Abstract: | This paper proves several extremal results for 3-connected matroids. In particular, it is shown that, for such a matroid M, (i) if the rank r(M) of M is at least six, then the circumference c(M) of M is at least six and, provided |E(M)|4r(M)−5, there is a circuit whose deletion from M leaves a 3-connected matroid; (ii) if r(M)4 and M has a basis B such that Me is not 3-connected for all e in E(M)−B, then |E(M)|3r(M)−4; and (iii) if M is minimally 3-connected but not hamiltonian, then |E(M)|3r(M)−c(M). |
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