Bondy's small cycle double cover conjecture

A cycle double cover of a graph is a list of cycles such that every edge belongs to exactly two of them. A graph is simple if it has no loops or parallel edges, and $2$-edge-connected if deleting any one edge leaves it connected.

Bondy's small cycle double cover conjecture. Every simple $2$-edge-connected graph on $n$ vertices has a cycle double cover consisting of at most $n-1$ cycles.

The bound is best possible for complete graphs. The source notes that this conjecture does not follow immediately from its proof of the cycle double cover theorem.