On graphs with α- and <em>b</em>-edge consecutive edge magic labelings

Abstract

<p>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 <em>n</em> is said to be <em>b</em>-edge consecutive when its edges are labeled with the integers <em>b+1</em>, <em>b+2</em>, ..., <em>b+n</em>, for some 0 ≤ <em>b</em> ≤ <em>n</em>. In this work, we prove the existence of several <em>b</em> edge-magic labelings for any graph of order <em>m</em> and size <em>m-1</em> that admits an α-labeling. In addition, we determine the exact value of <em>b</em> induced by the α-labeling, as well as for its reverse, complementary, and reverse complementary labelings.</p>