Geodesic
Orientable $ \mathbb{Z}_N $-Distance Magic Graphs
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 533-546

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Let G = (V, E) be a graph of order n. A distance magic labeling of G is a bijection 𝓁: V →1, 2, . . ., n for which there exists a positive integer k such that Σ_ x ∈ N(v) 𝓁 (x) = k for all v ∈ V, where N(v) is the open neighborhood of v. Tuttes flow conjectures are a major source of inspiration in graph theory. In this paper we ask when we can assign n distinct labels from the set 1, 2, . . ., n to the vertices of a graph G of order n such that the sum of the labels on heads minus the sum of the labels on tails is constant modulo n for each vertex of G. Therefore we generalize the notion of distance magic labeling for oriented graphs.
Keywords: distance magic graph, digraph, flow graph 
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     title = {Orientable $ \mathbb{Z}_N ${-Distance} {Magic} {Graphs}},
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Cichacz, Sylwia; Freyberg, Bryan; Froncek, Dalibor. Orientable $ \mathbb{Z}_N $-Distance Magic Graphs. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 533-546. http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a15/