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In this paper we deal with the problem of labeling the vertices, edges and faces of a toroidal and Klein bottle grid graphs with 4-sided faces by the consecutive integers from 1 up to + + and + + in such a way that the label of a 4-sided face and the labels of the vertices and edges surrounding that face all together add up to a weight of that face. These face-weights then form an arithmetic progression with common difference . The paper examines the existence of such labelings for several differences .