On rings of circuits in planar graphs

The terminology and notation of this paper will follow [1]. In that paper, it was conjectured that a graph is planar if and only if it has no maximal, odd ring of circuits. In fact, it was shown that every non-planar graph has such a ring, so that it remains only to demonstrate the non-existence of such rings in planar graphs. In this paper, we shall in fact produce a planar graph that contains a maximal, odd ring, thus refuting the conjecture. We then show how to modify the conjecture, and we prove that the revised conjecture is still satisfied by nonplanar graphs.