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. s a graph and C is a circuit in G , we say that two circuits in G form a split of C if the symmetric difference of their edges sets is equal to the edge set of C , and if they are separated in G by the intersection of their vertex sets. 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 C whenever G 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.