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