1‐rotational k ‐factorizations of the complete graph and new solutions to the Oberwolfach problem

We consider k ‐factorizations of the complete graph that are 1‐ rotational under an assigned group G , namely that admit G as an automorphism group acting sharply transitively on all but one vertex. After proving that the k ‐factors of such a factorization are pairwise isomorphic, we focus our attention to the special case of k  = 2, a case in which we prove that the involutions of G necessarily form a unique conjugacy class. We completely characterize, in particular, the 2‐factorizations that are 1‐rotational under a dihedral group. Finally, we get infinite new classes of previously unknown solutions to the Oberwolfach problem via some direct and recursive constructions. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 87–100, 2008