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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.

Summary. In the framework of cell-like membrane systems it is well known that the construction of exponential number of objects in polynomial time is not enough to efficiently solve NP-complete problems. Nonetheless, it may be sufficient to create an exponential number of membranes in polynomial time. In the framework of recognizer polarizationless P systems with active membranes, the construction of an exponential workspace expressed in terms of number of membranes and objects may not suffice to efficiently solve computationally hard problems.In this paper we study the computational efficiency of recognizer tissue P systems with communication (symport/antiport) rules and division rules. Some results have been already obtained in this direction: (a) using communication rules and forbidding division rules, only tractable problems can be efficiently solved; (b) using communication rules with length three and division rules, NP-complete problems can be efficiently solved. In this paper we show that the allowed length of communication rules plays a relevant role from the efficiency point of view of the systems.

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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.