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 $F$. Posed by G. Ringel in the 1960s, this problem is still open, even though infinitely many cases have been solved. For example, complete solutions are known for infinitely many prime orders $[3]$, when $F$ consists of cycles of the same length $[12]$, or when $F$ has two components $[21]$. In this paper, we give a complete solution of the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group $G$. Moreover, we characterize the infinite subgraphs $E$ of $F$ such that there exists a solution to $OP(F)$ containing a solution to $OP(E)$. 查看全文>>