Gracefulness of ∪form i=1 to n m_iC_4 ~2

The graceful graph is an important topic in graph theory. However, few studiers of the gracefulness of the disconnected graph have been made. The research work is especially inadequate on the gracefulness of n graphs' union. A kind of disconnected graph ∪ni=1m_iC~2_4 is presented,and it is proved that graph is both a graceful graph and an alternating graph, when m_i≥2, (i =1,2,…,n). And it is spreading widely.A method to make a kind of many graphs' union for the graceful graph and the alternating graph is given.