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Following standard terminology, $\lambda K_v$ is a multigraph on $v$ verticessuch that $\lambda$ edges join each pair of vertices. Let$(\lambda+\mu)K_{v+u}-\lambda K_v$ be the graph $(V\cup U,E)$ with $|V|=v$,$|U|=u$, and $(\lambda+\mu)-\lambda=\mu$ edges between the vertices $x$ and $y$if $x$ and $y$ both lie in $V$ and $\lambda+\mu$ edges between $x$ and $y$otherwise. The main result of this paper establishes necessary and sufficientconditions for an $m$-path decomposition of $(\lambda+\mu)K_{v+u}-\lambda K_v$.

Packing paths in a $(\lambda+\mu)K_{v+u}-\lambda K_v$