Cyclic Hamiltonian cycle systems of the λλ-fold complete and cocktail party graphs

We establish the necessary and sufficient conditions for the existence of a cyclic Hamiltonian cycle system of λKvλKv and of λ(K2v−I)λ(K2v−I). We also consider the existence problem for cyclic Hamiltonian cycle systems of λKvλKv that are simple  , i.e., without repeated cycles, solving it in the case that vv is a prime.