Note on Distance Magic Products $$G\circ C_4$$ G ∘ C 4

Abstract

A distance magic labeling of a graph $$G=(V,E)$$ of order $$n$$ is a bijection $$l :V \rightarrow \{1, 2,\ldots , n\}$$ with the property that there is a positive integer $$k$$ (called magic constant) such that $$w(x) = k$$ for every $$x \in V$$ . If a graph $$G$$ admits a distance magic labeling, then we say that $$G$$ is a distance magic graph. In the case of non-regular graph $$G$$ , the problem of determining whether there is a distance magic labeling of the lexicographic product $$G\circ C_4$$ was posted in Arumugam et al. (J Indonesian Math Soc 11–26, 2011). We give necessary and sufficient conditions for the graphs $$K_{m,n}\circ C_4$$ to be distance magic. We also show that the product $$C^{(t)}_3\circ C_4$$ of the Dutch Windmill Graph and the cycle $$C_4$$ is not distance magic for any $$t>1$$ .