Transforming Solutions for the Oberwolfach Problem into Solutions for the Spouse-Loving Variant

Abstract

The Oberwolfach problem $\mathrm{OP}\left(F\right)$, for a 2-factor $F$ of $K_n$, asks whether there exists a 2-factorization of $K_n$ (if $n$ is odd) or $K_n-I$ (if $n$ is even) where each 2-factor is isomorphic to $F$. Here, $I$ denotes any 1-factor of $K_n$. For even $n$, the problem OP$\left(F\right)$ may also be denoted ${\text{OP}}^{-}\left(F\right)$, and has been nicknamed the spouse-avoiding variant. Similarly, the spouse-loving variant is denoted ${\text{OP}}^{+}\left(F\right)$ and asks for a 2-factorization of $K_n+I$ (the complete graph with the edges of a 1-factor $I$ duplicated, rather than deleted) in which each 2-factor is isomorphic to $F$. To date, many more infinite families of cases of OP and ${\text{OP}}^{-}$ have been solved than of ${\text{OP}}^{+}\left(F\right)$. In this paper, we show how certain solutions to ${\text{OP}}^{-}\left(F\right)$ can be used to construct solutions to ${\text{OP}}^{+}\left(F\right)$; in particular, when the number of odd cycles in the 2-factor $F$ is not too large. Our technique of setups also allows us to completely solve the two-table ${\text{OP}}^{+}$; that is, ${\text{OP}}^{+}\left(F\right)$, where $F$ has exactly two components.