Elementary proofs of (relatively) recent characterizations of Eulerian graphs (Q1124607)

Toida showed that if a graph is Eulerian, then each edge belongs to an odd number of cycles. McKee established the converse of this implication using matroid theory. Shank used properties of binary vector spaces to show that a connected graph G is Eulerian if and only if the number of subsets of E(G) each of which is contained in a spanning tree of G, is odd. The author provides in this paper graph theoretical proofs of these results.