### On the existence of degree-magic labellings of the \(n\)-fold self-union of complete bipartite graphs

#### 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 \(1,2,\dots,|E(G)|\) 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.

#### Keywords

regular graphs, bipartite graphs, tripartite graphs, supermagic graphs, degree-magic graphs, balanced degree-magic graphs, magic rectangles

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