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{\bf Teresa Sousa}
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{\bf Minimum Weight $H$-Decompositions of Graphs: The Bipartite Case}
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Given graphs $G$ and $H$ and a positive number $b$, a \emph{weighted
$(H,b)$-decomposition} of $G$ is a partition of the edge set of $G$
such that each part is either a single edge or forms an
$H$-subgraph. We assign a weight of $b$ to each $H$-subgraph in the
decomposition and a weight of 1 to single edges. The total weight of
the decomposition is the sum of the weights of all elements in the
decomposition. Let $\phi(n,H,b)$ be the the smallest number such that
any graph $G$ of order $n$ admits an $(H,b)$-decomposition with weight
at most $\phi(n,H,b)$. The value of the function $\phi(n,H,b)$ when
$b=1$ was determined, for large $n$, by Pikhurko and Sousa
[\emph{Minimum {$H$}-Decompositions of Graphs}, Journal of
Combinatorial Theory, B, \textbf{97} (2007), 1041--1055.] Here we
determine the asymptotic value of $\phi(n,H,b)$ for any fixed
bipartite graph $H$ and any value of $b$ as $n$ tends to infinity.
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