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Splits of circuits. (English)
Discrete Math. 310, No. 21, 3026-3029 (2010).
Summary: 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 $G$ is 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. {\it E.E. García Moreno} and {\it T.R. Jensen}, A note on semiextensions of stable circuits, Discrete Math. 309, No. 15, 4952-4954 (2009; Zbl 1229.05167), 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.