On graphs with &#945;- and <em>b</em>-edge consecutive edge magic labelings

Abstract

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