Decompositions of multigraphs into parts with the same size

Given a family F of multigraphs without isolated vertices, a multigraph M is called F-decomposable if M is an edge disjoint union of multigraphs each of which is isomorphic to a member of F . We present necessary and sufficient conditions for existence of such decompositions if F consists of all multigraphs of size q except for one. Namely, for a multigraph H of size q we find each multigraph M of size kq, such that every partition of the edge set of M into parts of cardinality q contains a part which induces a submultigraph of M isomorphic to H .