Anti-magic labelling of Cartesian product of graphs

A graph G is anti-magic if there is a labelling of its edges with 1,2,…,|E| such that the sum of the labels assigned to edges incident to distinct vertices are different. In this paper, we prove that if G is k-regular for k≥2, then for any graph H with |E(H)|≥|V(H)|−1≥1, the Cartesian product H□G is anti-magic. We also show that if |E(H)|≥|V(H)|−1 and each connected component of H has a vertex of odd degree, or H has at least 2|V(H)|−2 edges, then the prism of H is anti-magic.