Totally magic graphs
A one-to-one mapping \(V\cup E \rightarrow \{1, 2, \ldots, |V|+|E|\}\) is vertex magic, if each vertex has the same sum of its label and the labels of its incident edges, and is edge magic if each edge has the same sum of its label and the labels of its endpoints. This paper investigates graphs that have a labelling that is both vertex magic and edge magic, i.e., totally magic. It appears that few graphs have such a labelling. The paper gives a number of necessary conditions, and a number of graphs with a totally magic labelling. Also, with help of computer analysis, the list all such graphs with up to 10 vertices is given.
scientific article; zbMATH DE number 1792453