### Abstract:

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, . . . , |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.