The 2-matching lattice of a graph (Q1124612)

Let L2M(G) be the lattice generated by the perfect 2-matchings of a graph G with \(2^ ab\), b odd, vertices, and let Lin 2M(G) be the linear subspace of \(R^ E\) generated by the perfect matchings of G. Clearly \(L2M(G)\leq Lin 2M(G).\) The result of this note is that a vector x in Lin 2M(G) is in L2M(G) iff the coordinate sum of x is divisible by \(2^ a\). This result is an analogy of a result of Lovász about perfect matchings in a graph.