Counting maximal cycles in binary matroids

AbstractIt is shown that each binary matroid contains an odd number of maximal cycles and, as a result of this, that each element of an Eulerian binary matroid is contained in an odd number of circuits

A proof of McKee's eulerian-bipartite characterization

AbstractA proof is given of the result about binary matroids that implies that a connected graph is Eulerian if and only if every edge lies in an odd number of circuits, and a graph is bipartite if and only if every edge lies in an odd number of cocircuits (minimal cutsets). A proof is also given of the result that the edge set of every graph can be expressed as a disjoint union of circuits and cocircuits. No matroid theory is assumed