The generalization of the Oberwolfach Problem, proposed by J. Liu in 2000, asks for a uniform $2$-factorization of the complete multipartite graph $K_{ m\times n}$. Here we focus our attention on $2$-factorizations regular under the cyclic group $Z_{ mn}$, whose $2$-factors are disjoint union of cycles all of even length $\ell$. In particular, we present a complete solution for the extremal cases $\ell=4$ and $\ell=mn$.
Categories: math.CO