On the super edge-magic deficiency of some graphs

A graph G is called super edge-magic if there exists a bijection [Formula: see text] , where [Formula: see text] , such that [Formula: see text] is a constant for every edge [Formula: see text]. Such a case, f is called a super edge magic labeling of G. A bipartite graph G with partite sets A and B is called consecutively super edge-magic if there exists a super edge-magic labeling f with the property that [Formula: see text] and [Formula: see text]. The super edge-magic deficiency of a graph G, denoted by [Formula: see text] , is either the minimum nonnegative integer n such that [Formula: see text] is super edge-magic or +∞ if there exists no such n. The consecutively super edge-magic deficiency of a bipartite graph G, denoted by [Formula: see text] , is either the minimum nonnegative integer n such that [Formula: see text] is consecutively super edge-magic or +∞ if there exists no such n. In this paper, we study the super edge-magic deficiency of some graphs. We investigate the (consecutively) super edge-magic deficiency of forests with two components. We also investigate the super edge-magic deficiency of a 2-regular graph [Formula: see text] and join product of [Formula: see text] with an isolated vertex.