Decompositions of infinite graphs: Part II circuit decompositions

We characterize the graphs that admit a decomposition into circuits, i.e. finite or infinite connected 2-regular graphs. Moreover, we show that, as is the case for the removal of a closed eulerian subgraph from a finite graph, removal of a non-dominated eulerian subgraph from a (finite or infinite) graph does not change its circuit-decomposability or circuit-indecomposability. For cycle-decomposable graphs, we show that in any end which contains at least n + 1 pairwise edge-disjoint rays, there are n edge-disjoint rays that can be removed from the graph without altering its cycle-decomposability. We also generalize the notion of the parity of the degree of a vertex to vertices of infinite degree, and in this way extend the well-known result that eulerian finite graphs are circuit-decomposable to graphs of arbitrary cardinality.