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.

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In this paper, we use the product ⊗ h in order to study super edge-magic labelings, bi-magic labelings and optimal k-equitable labelings. We establish, with the help of the product ⊗ h , new relations between super edge-magic labelings and optimal k-equitable labelings and between super edge-magic labelings and edge bi-magic labelings. We also introduce new families of graphs that are inspired by the family of generalized Petersen graphs. The concepts of super bi-magic and r -magic labelings are also introduced and discussed, and open problems are proposed for future research.2010 Mathematics subject classification: primary 05C78.

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