An Upper Bound for the Circumference of a 3-Connected Binary Matroid.

Jim Geelen and Peter Nelson proved that, for a loopless connected binary matroid M with an odd circuit, if a largest odd circuit of M has k elements, then a largest circuit of M has at most 2k - 2 elements. The goal of this note is to show that, when M is 3-connected, either M has a spanning circuit, or a largest circuit of M has at most 2k - 4 elements. Moreover, the latter holds when M is regular of rank at least four.