Vertex-magic total labelings of union of generalized Petersen graphs and union of special circulant graphs

Let G be a graph with vertex set V = V(G) and edge set E = E(G), and let n = \V(G)\ and e = \E(G)\. A vertex-magic total labeling (VMTL) of a graph is defined as a one-to-one mapping taking the vertices and edges onto the set of integers {1,2,..., n + e}, with the property that the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex. In this paper, we present the vertex magic total labeling of disjoint union of t generalized Petersen graphs U_j=1 ^tP(n_j,m_j), and disjoint union of t special circulant graphs U_j=1^tC_n(1,m_j).