D-Magic Oriented Graphs

Abstract

In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In particular, we label the vertices of the graph with distinct integers $\{1,2,\dots ,|V\left(G\right)\left|\right\}$ in such a way that the sum of all the vertex labels that are a distance in D away from a given vertex is the same across all vertices. We give some results related to the magic constant, construct a few infinite families of D-magic graphs, and examine trees, cycles, and multipartite graphs. This definition grew out of the definition of D-magic (undirected) graphs. This paper explores some of the symmetries we see between the undirected and directed version of D-magic labelings.