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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 \(t>1\).

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