When I use the term multigraph decomposition, I mean a partition of the edge set. In particular, a cycle decomposition of a multigraph is a partition of the edge set into cycles, where I am using cycle to indicate a connected subgraph in which each vertex has valency 2.

There is a short list of cycle decomposition problems that I view as important problems. At the top of my list is the so-called cycle double cover conjecture which is the underlying motivation for this book. My reasons for ranking it at the top are discussed next.

If a conjecture has been largely ignored, then longevity essentially is irrelevant, but when a conjecture has been subjected to considerable research, then longevity plays a significant role in its importance. The cycle double cover conjecture has been with us for more than thirty years and has received considerable attention including three special workshops devoted to just this single conjecture. Thus, just in terms of longevity the cycle double cover conjecture acquires importance.

There is a deep, but not well understood, connection with the structure of graphs for if a graph X contains no Petersen minor, then a vast generalization of the cycle double conjecture is true. Trying to understand what is going on in this realm adds considerably to the allure of the cycle double cover conjecture.

Another strong attraction of the conjecture is the connections with other subareas of graph theory. These include topological graph theory, graph coloring, and flows in graphs.