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 $K_v$ 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 $K_v$ 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.

令F为v阶的 2 正则图。奥伯沃尔法赫问题$○磷(F)$，于 1967 年提出并仍然开放，要求分解${ķ}_v$成F的副本。在本文中，我们表明$○磷(F)$只要F具有满足给定下限的足够大的循环，并且此外，具有单翻转自同构，它是一个对合自同构，它恰好反映了F的一个循环。此外，我们证明了问题的最小覆盖版本和最大打包版本的类似结果。当边缘${ķ}_v$有多重性 2，但在这种情况下，我们不要求F是单次翻转的。

我们的方法允许我们用表现良好的自同构显式构造 Oberwolfach 问题的解决方案，这与最近基于概率方法的一些渐近结果相反，这些结果是非构造性的，并且不提供F阶的下限来保证可解性的$○磷(F)$.

我们的构造基于一个加倍构造，该构造适用于删除顶点的 2 正则图的优雅标记。我们证明，只要路径分量的长度足够大，这类图是优雅的。对于存在的此类图的α标记，给出了路径长度的更好的下限。