On size, circumference and circuit removal in 3-connected matroids

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 M\e 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). © 2000 Elsevier Science B.V. All rights reserved

Even cycles in graphs

Let G be a 3-connected simple graph of minimum degree 4 on at least six vertices. The author proves the existence of an even cycle C in G such that G-V ( C ) is connected and G-E ( C ) is 2-connected. The result is related to previous results of Jackson, and Thomassen and Toft. Thomassen and Toft proved that G contains an induced cycle C such that both G-V ( C ) and G-E ( C ) is 2-connected. G does not in general contain an even cycle such that G-V ( C ) is 2-connected. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 163–223, 2004Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/34893/1/10156_ftp.pd

On removable circuits in graphs and matroids

Mader proved that every 2-connected simple graph G with minimum degree d exceeding three has a cycle C, the deletion of whose edges leaves a 2-connected graph. Jackson extended this by showing that C may be chosen to avoid any nominated edge of G and to have length at least d-1. This article proves an extension of Jackson\u27s theorem. In addition, a conjecture of Goddyn, van den Heuvel, and McGuinness is disproved when it is shown that a natural matroid dual of Mader\u27s theorem fails. © 1999 John Wiley & Sons, Inc