A non-existence result on cyclic cycle-decompositions of the cocktail party graph

We prove that in every cyclic cycle-decomposition of K2n−IK2n−I (the cocktail party graph of order 2n2n) the number of cycle-orbits of odd length must have the same parity of n(n−1)/2n(n−1)/2. This gives, as corollaries, some useful non-existence results one of which allows to determine when the two table Oberwolfach Problem OP(3,2ℓ)OP(3,2ℓ) admits a 1-rotational solution.