Splits of circuits

This paper discusses an attempt at identifying a property of circuits in (nonplanar) graphs resembling the separation property of circuits in planar graphs derived from the Jordan Curve Theorem.If GG is a graph and CC is a circuit in GG, we say that two circuits in GG form a split of CC if the symmetric difference of their edges sets is equal to the edge set of CC, and if they are separated in GG by the intersection of their vertex sets.García Moreno and Jensen, A note on semiextensions of stable circuits, Discrete Math. 309 (2009) 4952–4954, asked whether such a split exists for any circuit CC whenever GG is 3-connected. We observe that if true, this implies a strong form of a version of the Cycle Double-Cover Conjecture suggested in the Ph.D. thesis of Luis Goddyn. The main result of the paper shows that the property holds for Hamilton circuits in cubic graphs.