  Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem,
$OP(F)$, asks for a $2$-factorization of the complete graph on $v$ vertices in
which each $2$-factor is isomorphic to $F$. In this paper, we give a complete
solution to the Oberwolfach problem over infinite complete graphs, proving the
existence of solutions that are regular under the action of a given involution
free group $G$. We will also consider the same problem in the more general
contest of graphs $F$ that are spanning subgraphs of an infinite complete graph
$\mathbb{K}$ and we provide a solution when $F$ is locally finite. Moreover, we
characterize the infinite subgraphs $L$ of $F$ such that there exists a
solution to $OP(F)$ containing a solution to $OP(L)$.

Combinatorics