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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 regularunder the cyclic group $Z_{mn}$, whose $2$-factors are disjoint union of cyclesall of even length $\ell$. In particular, we present a complete solution forthe extremal cases $\ell=4$ and $\ell=mn$.

Cyclic uniform 2-factorizations of the complete multipartite graph