Regular Oberwolfach problems and group sequencings

摘要

We deal with Oberwolfach factorizations of the complete graphs K-n 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 K-n (n odd) which admits a regular group of automorphisms G (G = (n - 1)/2) exists if and only if n equivalent to 3 (mod 4), without further restrictions on the structure of G. (C) 2001 Academic Press. [References: 18]