Some bounds on the number of non isomorphic cyclic k−cycle systems of the complete graph

In a recent paper M. Buratti and M. E. Muzychuk have established some lower bounds on the number of non isomorphic cyclic Steiner Triple Systems of order v ≡ 1 (mod 6). We complete their results to the case v ≡ 3 (mod 6). Furthermore, we find lower bounds for the number of non isomorphic cyclic (K_ (v), C_ (k) ) -designs for each admissible pair (v, k). We place somewhat emphasis on the case k odd and v ≡ 1, k (mod 2k). We also find lower bounds for the number of cyclic Hamiltonian cycle systems of order 3p, p > 3 a prime.