Transforming solutions for the Oberwolfach problem into solutions for the spouse-loving variantLekše, Maruša (Avtor) Šajna, Mateja (Avtor) 1‐rotational solution12‐setup2‐factorization2‐starter6‐setupcomplete graph plus a 1‐factorOberwolfach problemspouse‐loving variantThe Oberwolfach problem ${\mathrm {OP}}(F)$, 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 ${\mathrm {OP}}(F)$ may also be denoted ${\mathrm {OP}}^-(F)$, and has been nicknamed the spouse-avoiding variant. Similarly, the spouse-loving variant is denoted ${\mathrm {OP}}^+(F)$ 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 ${\mathrm {OP}}$ and ${\mathrm {OP}}^-$ have been solved than of ${\mathrm {OP}}^+(F)$. In this paper, we show how certain solutions to ${\mathrm {OP}}^-(F)$ can be used to construct solutions to ${\mathrm {OP}}^+(F)$; 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 ${\mathrm {OP}}^+$; that is, ${\mathrm {OP}}^+(F)$, where $F$ has exactly two components.20262026-07-23 12:37:25Neznano31266UDK: 519.17ISSN pri članku: 1063-8539DOI: 10.1002/jcd.70020COBISS_ID: 285786371sl