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Orientable \(\mathbb{Z}_N\)-distance magic graphs. (English) Zbl 1404.05185
Summary: Let \(G = (V, E)\) be a graph of order \(n\). A distance magic labeling of \(G\) is a bijection \(\ell: V\rightarrow\{1, 2, \dots, n\}\) for which there exists a positive integer \(k\) such that \(\sum_{x\in N(v)}\ell(x) = k\) for all \(v\in V\), 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 \(\{1, 2, \dots, n\}\) 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.
MSC:
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C15 Coloring of graphs and hypergraphs
05C22 Signed and weighted graphs
05C20 Directed graphs (digraphs), tournaments
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C76 Graph operations (line graphs, products, etc.)
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