The binary matroids whose only odd circuits are triangles

This paper generalizes a graph-theoretical result of Maffray to binary matroids. In particular, we prove that a connected simple binary matroid M has no odd circuits other than triangles if and only if M is affine, M   is M(K4)M(K4) or F7F7, or M is the cycle matroid of a graph consisting of a collection of triangles all of which share a common edge. This result implies that a 2-connected loopless graph G has no odd bonds of size at least five if and only if G is Eulerian or G   is a subdivision of either K4K4 or the graph that is obtained from a cycle of parallel pairs by deleting a single edge.