The distance magic property and two families of Cartesian product graphs - Headley, Patrick Thomas (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2025)
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 Σy ∈ N(x)f(y)=s for all x ∈ V. In this paper we show that the only distance magic graph of the form Kn□Cm is K1□C4, and that m = 4 if Cm□Kn, t is distance magic. Necessary conditions are given for C4□Kn, 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□Kn, t with n > t, in particular constructing an infinite sequence of non-isomorphic distance magic graphs of this type.