On the Oberwolfach problem for single-flip 2-factors via graceful labelings

Let F be a 2-regular graph of order v. The Oberwolfach problem OP(F), posed in 1967 and still open, asks for a decomposition of Kv into copies of F. In this paper we show that OP(F) has a solution whenever F has a sufficiently large cycle which meets a given lower bound and, in addition, has a single-flip automorphism, which is an involutory automorphism acting as a reflection on exactly one of the cycles of F. Furthermore, we prove analogous results for the minimum covering version and the maximum packing version of the problem. We also show a similar result when the edges of Kv have multiplicity 2, but in this case we do not require that F be single-flip. Our approach allows us to explicitly construct solutions to the Oberwolfach Problem with well-behaved automorphisms, in contrast with some recent asymptotic results, based on probabilistic methods, which are nonconstructive and do not provide a lower bound on the order of F that guarantees the solvability of OP(F). Our constructions are based on a doubling construction which applies to graceful labelings of 2-regular graphs with a vertex removed. We show that this class of graphs is graceful as long as the length of the path-component is sufficiently large. A much better lower bound on the length of the path is given for an α-labeling of such graphs to exist.