  A graph $G$ is called edge-magic if there is a bijective function $f$ from
the set of vertices and edges to the set $\{1,2,\ldots,|V(G)|+|E(G)|\}$ such
that the sum $f(x)+f(xy)+f(y)$ for any $xy$ in $E(G)$ is constant. Such a
function is called an edge-magic labelling of G and the constant is called the
valence of $f$. An edge-magic labelling with the extra property that $f(V(G))=
\{1,2,\ldots,|V(G)|\}$ is called super edge-magic. In this paper, we establish
a relationship between the valences of (super) edge-magic labelings of certain
types of bipartite graphs and the existence of a particular type of
decompositions of such graphs.
