Orientable \mathbb{Z}_{n}-distance magic graphs

Abstract

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 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.

Distance magic-type and distance antimagic-type labelings of graphs

“…The following analog of group distance magic labeling for directed graphs was introduced in [14]. Let G = (V, E) be a graph.…”

“…In this chapter we study a generalization of distance magic graphs introduced recently in [14]. Let G be a simple, undirected graph on n vertices.…”

“…We now consider the analogous labeling in the setting of directed graphs first introduced in [14]. We begin with an undirected graph G = (V, E).…”

“…Let Z n be the cyclic group of order n. With regards to orientable Z n -distance magic labeling, Cichacz et. al characterized complete graphs, complete bipartite graphs, complete tripartite graphs, circulant graphs, and certain products of graphs in [14].…”

“…The motivation for our study is a conjecture stated in their concluding remarks [14]. Determining whether an arbitrary graph is orientable Z n -distance magic is not practical.…”