Cycle decompositions and double covers of graphs

One of the earliest results in graph theory is related to the characterization of graphs that have cycle decompositions: a graph G has a cycle decomposition if and only if every vertex of G has even degree.  One way of generalizing the notion of a cycle decomposition to a graph that has vertices of odd degree is a cycle double cover.  The obvious condition that is necessary for a graph to have a cycle double cover is that the graph be bridgeless, and it has been conjectured that this condition is also sufficient.  My interest is in a refinement of this conjecture: every simple bridgeless graph on n vertices has a cycle double cover with at most (n-1) cycles.  I will discuss some progress that has been made on this conjecture, and also an analogous conjecture regarding the minimum number of cycles in a cycle decomposition of a connected even graph.