There is some hint that 2 may be special with respect to the partition result proved above. Namely, if 2 divides the number of edges of Knand the valence is even, that is, , there is a partition of the edge set of Kn into two isomorphic circulants if and only if all primes dividing n are congruent to 1 modulo 4.
Is there some extension of the result to other k>2. If k>2, it is easy to construct partitions of the edge set of Kn into kisomorphic circulant graphs if every prime dividing n is congruent to 1 modulo 2k. However, this condition is no longer necessary. For example, the edges of K15 can be partitioned into seven Hamilton cycles. They are certainly circulant graphs, but 15 is not a product of primes congruent to 1 modulo 14.
This suggests an obvious question. One also may want to require that the circulant subgraphs span Kn.