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