Decomposing complete equipartite multigraphs into cycles of variable lengths: the amalgamation-detachment approach

Using the technique of amalgamation-detachment, we show that the complete equipartite multigraph $\lambda K_{n\times m}$ can be decomposed into cycles of lengths $c_{1}m,\dots ,c_{k}m$ (plus a 1-factor if the degree is odd) whenever there exists a decomposition of $\lambda m{K}_n$ into cycles of lengths $c_1, \dots,c_k$ (plus a 1-factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph $\lambda K_{n\times m}$.