Packing paths in a $(λ+μ)K_{v+u}-λK_v$

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

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