Cyclic Uniform 2-Factorizations of the Complete Multipartite Graph

A generalization of the Oberwolfach problem, proposed by Liu (J Comb Des 8:42–49, 2000 ), asks for a uniform 2-factorization of the complete multipartite graph K m × n . Here we focus our attention on cyclic 2-factorizations, whose 2-factors are disjoint union of cycles all of even length ℓ . In particular, we present a complete solution for the extremal cases ℓ = 4 and ℓ = m n .