Efficient Subgraph Packing

We are concerned with a natural generalization of the matching problem: the packing of graphs from a pregiven family. We characterize completely the complexity of deciding the existence of a perfect packing, if the pregiven family consists of two graphs, one of which is one edge on two vertices. Our results complete the work of Cornuejols. Hartvigsen, and Pulleyblank, and Hell and Kirkpatrick.