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 k=2, a case in which we prove 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 prviously unknown solutions to the Oberwolfach problem via some direct and recursive constructions.