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Path And Cycle Decompositions

First I want to say a few words about my graph terminology. If I want to allow loops, I use the adjective reflexive. If I want to allow multiple edges, I use multigraph. Thus, a graph has no loops and no multiple edges. I use valency rather than degree. If we say a graph is 4-valent (or tetravalent), it means it is regular of valency 4, for example. A decomposition of a graph X is a partition of its edge set into subgraphs. There are two typical situations. Either we want all the subgraphs to be isomorphic to some fixed graph Y . We shall call this a Y -decomposition of X, or a decomposition of X into subgraphs isomorphic to Y . The other typical situation is that we are given a list L of subgraphs and we want a 1-1 correspondence between the parts of the decomposition and the members of L.