Abstract
be a matroid with a maximum-sized circuit C of size at least four. This paper proves that, for k∈{2,3}, if M is k-connected, then every element of M is contained in a circuit of size at least \(\). Even when M is 3-connected and binary, the presence of a large circuit in M does not guarantee that M has a large circuit containing a nominated pair of elements. However, when M is 3-connected and graphic, it will be shown that every pair of distinct elements is contained in a circuit of at least \(\). Examples will be given to show that these results are best-possible and some related results will also be proved.
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