Distance magic labelings of graphs

As a natural extension of previously defined graph labelings, we introduce in this paper a new magic labeling whose evaluation is based on the neighbourhood of a vertex. We define a 1-vertex-magic vertex labeling of a graph with v vertices as a bijection f taking the vertices to the integers 1, 2, . . . , v with the property that there is a constant k such that at any vertex x, ∑ y∈N(x) f(y) = k, where N(x) is the set of vertices adjacent to x. We completely solve the existence problem of 1-vertex-magic vertex labelings for all complete bipartite, tripartite and regular multipartite graphs, and obtain some non-existence results for other natural families of graphs.