On the super edge-magic deficiency of some graphs

A graph G is called super edge-magic if there exists a bijection f:V(G)∪E(G)⟶{1,2,⋯,|V(G)|+|E(G)|}, where f(V(G))={1,2,⋯,|V(G)|}, such that f(u)+f(uv)+f(v) is a constant for every edge uv∈E(G). 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 f(A)={1,2,⋯,|A|} and f(B)={|A|+1,|A|+2,⋯,|V(G)|}. The super edge-magic deficiency of a graph G, denoted by μs(G), is either the minimum nonnegative integer n such that G∪nK1 is super edge-magic or +∞ if there exists no such n. The consecutively super edge-magic deficiency of a bipartite graph G, denoted by μc(G), is either the minimum nonnegative integer n such that G∪nK1 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 2C3∪Cn and join product of K1,n∪Pm with an isolated vertex.
(Consecutively) super edge-magic graph; (Consecutively) super edge-magic deficiency; Forest with two components; 2-regular graph; Join product graph