A graph partition problem

Given a graph G on n vertices, for which m is it possible to partition the edge set of the m-fold complete graph mK_n into copies of G? We show that there is an integer m₀, which we call the partition modulus of G, such that the set M(G) of values of m for which such a partition exists consists of all but finitely many multiples of m₀. Trivial divisibility conditions derived from G give an integer m₁ that divides m₀; we call the quotient m₀/m₁ the partition index of G. It seems that most graphs G have partition index equal to 1, but we give two infinite families of graphs for which this is not true. We also compute M(G) for various graphs and outline some connections between our problem and the existence of designs of various types.