Two graphs and , each on vertices, pack if there exists a bijection from onto such that only if . In 2014, Alon and Yuster proved that, for sufficiently large , if and , then and pack. In this paper, we characterize the pairs of graphs for which the theorem of Alon and Yuster is sharp. We also prove the stronger result that for sufficiently large , if , , and , then and pack whenever there is a vertex such that and .