For any non-trivial abelian group A, a graph G is said to be A-magic if there exists a labeling f of the edges of G with non zero elements of A such that the vertex labeling f + defned as f +(v) = Σf(uv) taken over all edges uv incident at v is a constant4. A graph is said to be A-magic if it admits A- magic labeling. In this paper we consider (moduloZ4,+) as abelian group and we prove Z4 - magic labeling for various graphs and generalize Z4p -magic labeling for those graphs. The graphs which admit Z4p-magic labeling are called as Z4p-magic graphs.