Note on Distance Magic Products G ∘ C4

A distance magic labeling of a graph G = (V, E) of order n is a bijection l : V → {1, 2, . . . , n} with the property that there is a positive integer k (called magic constant) such that w(x) = k for every x ∈ 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 ◦C4 was posted in Arumugam et al. (J Indonesian Math Soc 11–26, 2011). We give necessary and sufficient conditions for the graphs Km,n ◦ C4 to be distance magic. We also show that the product C (t) 3 ◦ C4 of the Dutch Windmill Graph and the cycle C4 is not distance magic for any t > 1.