On V-super and E-super vertex-magic total labelings of graphs

Let G be a graph of order p and size q. A vertex-magic total labeling is an assignment of the integers 1,2,…,p+q to the vertices and the edges of G, so that at each vertex, the vertex label and the labels on the edges incident at that vertex, add to a fixed constant, called the magic constant of G. Such a labeling is V-super vertex-magic total if f(V(G))={1,2,…,p} and is E-super vertex-magic total if f(E(G))={1,2,…,q}. A graph that admits a V-super vertex-magic total labeling is called V-super vertex-magic. Similarly, a graph that admits an E-super vertex-magic total labeling is called E-super vertex-magic. In this paper, we prove a necessary and sufficient condition for the existence of V-super vertex-magic labeling. Further, we find V-super vertex-magic total labeling and E-super vertex-magic total labeling of some families of graphs.