Regular Oberwolfach Problems and Group Sequencings

We deal with Oberwolfach factorizations of the complete graphs Kn and K*n, which admit a regular group of automorphisms. We show that the existence of such a factorization is equivalent to the existence of a certain difference sequence defined on the elements of the automorphism group, or to a certain sequencing of the elements of that group. In the particular case of a hamiltonian factorization of the directed graph K*n which admits a regular group of automorphisms G (|G|=n−1), we have that such a factorization exists if and only if G is sequenceable. We shall demonstrate how the mentioned above (difference) sequences may be used in the construction of such factorizations. We prove also that a hamiltonian factorization of the undirected graph Kn (n odd) which admits a regular group of automorphisms G (|G|=(n−1)/2) exists if and only if n≡3 (mod 4), without further restrictions on the structure of G.