New problems related to the valences of (super) edge-magic labelings

A graph G of order p and size q is edge-magic if there is a bijective function f : V (G) ∪ E(G) −→ {i} p+q i=1 such that f(x) + f(xy) + f(y) = k , for all xy ∈ E(G) . The function f is an edge-magic labeling of G and the sum k is called either the magic sum, the valence or the weight of f . Furthermore, if f(V (G)) = {i} p i=1 then f is a super edge-magic labeling of G . In this paper we study the valences that can be attained by (super) edge-magic labelings of some families of graphs.The first and the third author are supported by the Spanish Research Council under project MTM2011-28800-C02-01 and by the Catalan Research Council under grant 2009SGR1387