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Let $\\lambda K_{v,u}$ be the graph $(V\\cup U,E)$ with $|V|=v$, $|U|=u$, and \\[ \\mu_G(xy)=\\begin{cases} \\lambda &\\mbox{if $x\\in U$ and $y\\in V$ or if $x\\in V$ and $y\\in U$}\\\\ 0 &\\mbox{otherwise.} \\\\ \\end{cases} \\] Let $M$ be a sequence of non-negative integers $m_1,m_2,\\ldots,m_n$. An $(M)$-cycle decomposition of a graph $G$ is a partition of the edge set into cycles of lengths $m_1,m_2,\\ldots,m_n$. 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