Cyclic uniform 2-factorizations of the complete multipartite graph

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 $\mathrm{\ell }$. In particular, we present a complete solution for the extremal cases $\mathrm{\ell }=4$ and $\mathrm{\ell }=mn$.