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

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