A Problem on Edge-magic Labelings of Cycles

Abstract

AbstractIn 1970, Kotzig and Rosa defined 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 : V(G)⊔E(G) → {1; 2, … p+q} is an edge-magic labeling of G if f (u)+ f (uv)+ f (v) = k, for all uv ∊ E(G). 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 states that all possible theoretical magic sums are attained for each cycle of order n ≥ 7. Motivated by this conjecture, we prove that for all n0∊ N, there exists n 2 N such that the cycle Cn admits at least n0 edge-magic labelings with at least n0 mutually distinct magic sums. We do this by providing a lower bound for the number of magic sums of the cycle Cn, depending on the sum of the exponents of the odd primes appearing in the prime factorization of n.