A note on semiextensions of stable circuits

A semiextension of a circuit C in a graph G provides a possible means of finding a cycle double cover of G with C as a prescribed circuit. Recently we conjectured [E.E. García Moreno, T.R. Jensen, On semiextensions and circuit double covers, J. Combin. Theory Ser. B 97 (2007) 474–482] that if G is cubic and 2-edge-connected, then a semiextension of C in G exists. If true, this would imply several long-standing conjectures. re is a circuit C ′ in G with C ′ ≠ C and V ( C ) ⊆ V ( C ′ ) , then C ′ is called an extension of C , a special case of a semiextension. If there is no such circuit, then C is said to be stable in G . Hence the existence question for semiextensions is easy except for stable circuits. Not many examples of graphs with stable circuits have been published. In this note we show that the members of a particular class of stable circuits described by M. Kochol have semiextensions.