(Di)graph decompositions and magic type labelings: a dual relation

A graph $G$ is called edge-magic if there is a bijective function $f$ from the set of vertices and edges to the set $\{1,2,\dots ,\mathrm{\mid }V(G)\mathrm{\mid }+\mathrm{\mid }E(G)\mathrm{\mid }\}$ 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.