#### Title

The binary matroids whose only odd circuits are triangles

#### Document Type

Article

#### Publication Date

5-1-2016

#### Abstract

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.

#### Publication Source (Journal or Book title)

Advances in Applied Mathematics

#### First Page

34

#### Last Page

38

#### Recommended Citation

Oxley, J., & Wetzler, K.
(2016). The binary matroids whose only odd circuits are triangles.* Advances in Applied Mathematics**, 76*, 34-38.
https://doi.org/10.1016/j.aam.2016.01.006