Perfect (super) Edge-Magic Crowns

A graph G is called edge-magic if there is a bijective function f from the set of vertices and edges to the set {1, 2, ...,vertical bar V(G)vertical bar + vertical bar E(G)vertical bar} such that the sum f (x) + f(xy) + f(y) for any xy in E(G) is constant. Such a function is called an edge-magic labelling of G and the constant is called the valence. An edge-magic labelling with the extra property that f(V(G)) = {1, 2, ..., vertical bar V (G)vertical bar} is called super edge-magic. A graph is called perfect (super) edge-magic if all theoretical (super) edge-magic valences are possible. In this paper we continue the study of the valences for (super) edge-magic labelings of crowns C-m circle dot (K) over bar (n) and we prove that the crowns are perfect (super) edge-magic when m = pq where p and q are different odd primes. We also provide a lower bound for the number of different valences of C-m circle dot (K) over bar (n), in terms of the prime factors of m.