Concluding Remarks

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 K_nand the valence is even, that is, $n\equiv 1\mbox{(mod }4)$, there is a partition of the edge set of K_n 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 K_n 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 K_n.