Approximate Cycle Double Cover

The Cycle double cover (CDC) conjecture states that for every bridgeless graph G, there exists a family \(\mathcal {F}\) of cycles such that each edge of the graph is contained in exactly two members of \(\mathcal {F}\) . Given an embedding of a graph G, an edge e is called a bad edge if it is visited twice by the boundary of one face. The CDC conjecture is equivalent to bridgeless cubic graphs having an embedding with no bad edge. In this work, we introduce nontrivial upper bounds on the minimum number of bad edges in an embedding of a cubic graph. Moreover, we present efficient algorithms to find embeddings satisfying these bounds.