A characterization of planar cubic graphs

If S is a collection of circuits in a graph G, the circuits in S are said to be consistently orientable if G can be oriented so that they are all directed circuits. If S is a set of three or more consistently orientable circuits such that no edge of G belongs to more than two circuits of S, then S is called a ring if there exists a cyclic ordering C₀, C₁,…, C_{n ? 1}, C₀ of the n circuits in S such that $\text{EC}{}_{\mathrm{i}}\text{? EC}{}_{\mathrm{j}}\text{≠ ?}$ if and only if j = i or j ≡ i ? 1 (mod n) or j ≡ i + 1 (mod n). We characterise planar cubic graphs in terms of the non-existence of a ring with certain specified properties.