The place of super edge-magic labelings among other classes of labelings Original Research Article

A (p,q)-graph G is edge-magic if there exists a bijective function f:V(G)∪E(G)→{1,2,…,p+q} such that f(u)+f(v)+f(uv)=k is a constant, called the valence of f, for any edge uv of G. Moreover, G is said to be super edge-magic if f(V(G))={1,2,…,p}. In this paper, we present some necessary conditions for a graph to be super edge-magic. By means of these, we study the super edge-magic properties of certain classes of graphs. We also exhibit the relationships between super edge-magic labelings and other well-studied classes of labelings. In particular, we prove that every super edge-magic (p,q)-graph is harmonious and sequential (for a tree or q⩾p) as well as it is cordial, and sometimes graceful. Finally, we provide a closed formula for the number of super edge-magic graphs.