C.-Q. Zhang

*The second circuit conjecture and cycle double cover conjecture*

**Abstract**. The Cycle Double Cover Conjecture (CDC) was proposed independently by P.D. Seymour (1979) and G. Szekeres (1973). The conjecture is easy to state: For every 2-connected graph, there is a family of circuits such that every edge is contained in precisely two members of the family. This talk will survey some approaches to this well-known open problem in graph theory.

It was asked by Seymour that, for every cubic, bridgeless graph *G* and every circuit *C* of *G*, whether or not *G* contains a circuit *C'* distinct from *C* with *V*(*C* ) ⊆ *V*(*C'*) (The Second Circuit Problem).

This problem, if true, implies the famous cycle double cover conjecture. Although a counterexample was discovered by Fleischner (1994), the Second Circuit Problem remains as a promising approach to a CDC conjecture.

In this talk, we will survey some old and recent results, and propose some modifications of this problem and possible approaches to CDC conjecture