Geodesic
On the existence of degree-magic labellings of the $n$-fold self-union of complete bipartite graphs
Algebra and discrete mathematics, Tome 28 (2019) no. 1, pp. 107-122

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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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     title = {On the existence of degree-magic labellings of the $n$-fold self-union of complete bipartite graphs},
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     url = {http://geodesic.mathdoc.fr/item/ADM_2019_28_1_a7/}
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Phaisatcha Inpoonjai; Thiradet Jiarasuksakun. On the existence of degree-magic labellings of the $n$-fold self-union of complete bipartite graphs. Algebra and discrete mathematics, Tome 28 (2019) no. 1, pp. 107-122. http://geodesic.mathdoc.fr/item/ADM_2019_28_1_a7/