The distance magic property and two families of Cartesian product graphs

Abstract

Let G = G(V, E) be a simple graph. The graph G is said to be distance magic if there exists a bijection f : V → {1, 2, …, |V|} and a constant s such that Σ_{y ∈ N(x)}f(y)=s for all x ∈ V. In this paper we show that the only distance magic graph of the form K_n□C_m is K₁□C₄, and that m = 4 if C_m□K_n, t is distance magic. Necessary conditions are given for C₄□K_n, t to be distance magic when n > t. These conditions are shown to be sufficient when n and t are both even. We conclude with some examples of distance magic graphs of the form C₄□K_n, t with n > t, in particular constructing an infinite sequence of non-isomorphic distance magic graphs of this type.