The binary matroids whose only odd circuits are triangles

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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) or F7, 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 K4 or the graph that is obtained from a cycle of parallel pairs by deleting a single edge.

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Advances in Applied Mathematics

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