The Cycle Double Cover Conjecture A cycle in a graph is a simple closed walk. The following Double Cover Conjecture is one of the most famous problems in graph theory.
Every bridgeless graph has a collection of cycles which together contain every edge exactly twice.
The conjecture is almost easy. However, some of these cycles may be of length two and hence do not correspond to cycles in a double cover of G . A stumbling block to inductive proofs has been found in many different contexts. There are several variations of a topological nature. For example there is the Circular Embedding Conjecture.