On graphs with α- and b-edge consecutive edge magic labelings

Among the most studied graph labelings we have the varieties called alpha and edge-magic. Even when their definitions seem completely different, these labelings are related. A graceful labeling of a bipartite graph is called an α-labeling if the smaller labels are assigned to vertices of the same stable set. An edge-magic labeling of a graph of size n is said to be b-edge consecutive when its edges are labeled with the integers b+1, b+2, ..., b+n, for some 0 ≤ b ≤ n. In this work, we prove the existence of several b edge-magic labelings for any graph of order m and size m-1 that admits an α-labeling. In addition, we determine the exact value of b induced by the α-labeling, as well as for its reverse, complementary, and reverse complementary labelings.