A survey of the cycle double cover conjecture

This conjecture has been attributed variously to many mathematicians, but was known to be a consequence of the Strong Embedding Conjecture (Conjecture 2) by W. Tutte, G. Haggard, as well as by G. Szekeres and P. Seymour, amongst others [18], [16]. In what follows, we survey some of what is known about the above conjecture and discuss various related problems and techniques. We discuss the strong embedding conjecture and some variants in Section 2. We consider the structure of a minimal counterexample in Section 3. In Sections 4 and 5, we discuss generalizations to cycle k-covers and to integer combinations of cycles, respectively. In preparing this document, the author found F. Jaeger’s survey article [12] and to C.Q. Zhang’s book [20] to be extremely helpful. All of our graphs may contain loops and multiple edges.