Circuit extension and circuit double cover of graphs

Let G be a cubic graph and C be a circuit. An extension of C is a circuit D such that V ( C ) ⊆ V ( D ) and E ( C ) ≠ E ( D ) . The study of circuit extension is motivated by the circuit double cover conjecture. It is proved by Fleischner (1990) that a circuit C is extendable if C has only one non-trivial Tutte bridge. It is further improved by Chan, Chudnovsky and Seymour (2009) that a circuit is extendable if it has only one odd Tutte bridge. Those earlier results are improved in this paper that C is extendable if all odd Tutte bridges of C are sequentially lined up along C . It was proved that if every circuit is extendable for every bridgeless cubic graph, then the circuit double cover conjecture is true (Kahn, Robertson, Seymour 1987). Although graphs with stable circuits have been discovered by Fleischner (1994) and Kochol (2001), variations of this approach remain one of most promising approaches to the circuit double cover conjecture. Following some early investigation of Seymour and Fleischner, we further study the relation between circuit extension and circuit double cover conjecture, and propose a new approach to the conjecture. This new approach is verified for some graphs with stable circuits constructed by Fleischner and Kochol.