  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 $m_0$, which we call the \emph{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 $m_0$. Trivial
divisibility conditions derived from $G$ give an integer $m_1$ which divides
$m_0$; we call the quotient $m_0/m_1$ the \emph{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.
