On the existence of degree-magic labellings of the $n$-fold self-union of complete bipartite graphs

Magic rectangles are a classical generalization of the well-known magic squares, and they are related to graphs. A graph $G$ is called degree-magic if there exists a labelling of the edges by integers $1,2,\dots,|E(G)|$ such that the sum of the labels of the edges incident with any vertex $v$ is equal to $(1+|E(G)|)\deg(v)/2$. Degree-magic graphs extend supermagic regular graphs. In this paper, we present a general proof of the necessary and sufficient conditions for the existence of degree-magic labellings of the $n$-fold self-union of complete bipartite graphs. We apply this existence to construct supermagic regular graphs and to identify the sufficient condition for even $n$-tuple magic rectangles to exist.