Decomposition of a complete bipartite multigraph into arbitrary cycle sizes

In a graph $G$, let ${\mu }_G(xy)$ denote the number of edges between $x$ and $y$ in $G$. Let $\lambda K_{v,u}$ be the graph $(V\cup U,E)$ with $\mathrm{\mid }V\mathrm{\mid }=v$, $\mathrm{\mid }U\mathrm{\mid }=u$, and \mu_G(xy)=\begin{cases} \lambda &if $x\in U$ and $y\in V$ or if $x\in V$ and $y\in U$\\ 0 &otherwise. \\ \end{cases} Let $M$ be a sequence of non-negative integers $m_1,m_2,\dots ,m_n$. An $(M)$-cycle decomposition of a graph $G$ is a partition of the edge set into cycles of lengths $m_1,m_2,\dots ,m_n$. In this paper, we establish necessary and sufficient conditions for the existence of an $(M)$-cycle decomposition of $\lambda K_{v,u}$.