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.
Abstract. Kotzig and Rosa defined in 1970 the concept of edge-magic labelings as follows: let G be a simple (p, q)-graph (that is, a graph of order p and size q without loops or multiple edges). A bijective function f :A graph that admits an edge-magic labeling is called an edge-magic graph, and k is called the magic sum of the labeling. An old conjecture of Godbold and Slater sets that all possible theoretical magic sums are attained for each cycle of order n ≥ 7. Motivated by this conjecture, we prove that for all n 0 ∈ N, there exists n ∈ N, such that the cycle C n admits at least n 0 edge-magic labelings with at least n 0 mutually distinct magic sums. We do this by providing a lower bound for the number of magic sums of the cycle C n , depending on the sum of the exponents of the odd primes appearing in the prime factorization of n.
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