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

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