On b-edge consecutive edge labeling of some regular tree

Abstract

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Let </span><span><em>G</em> </span><span>= (</span><span><em>V</em>,<em> E</em></span><span>) </span><span>be a finite (non-empty), simple, connected and undirected graph, where </span><span>V </span><span>and </span><span>E </span><span>are the sets of vertices and edges of </span><span>G</span><span>. An edge magic total labeling is a bijection </span><span>α </span><span>from </span><span><em>V</em> </span><span>∪ </span><span><em>E</em> </span><span>to the integers </span><span>1</span><span>, </span><span>2</span><span>, . . . , <em>n</em> </span><span>+ </span><em>e</em><span>, with the property that for every </span><span><em>xy</em> </span><span>∈ </span><em>E</em><span>, </span><span>α</span><span>(</span><em>x</em><span>) + </span><span>α</span><span>(</span><em>y</em><span>) + </span><span>α</span><span>(</span><em>xy</em><span>) = </span><em>k</em><span>, for some constant </span><em>k</em><span>. Such a labeling is called a </span><em>b</em><span>-edge consecutive edge magic total if </span><span>α</span><span>(</span><em>E</em><span>) = </span><span>{</span><span><em>b</em> </span><span>+ 1</span><span>, <em>b</em> </span><span>+ 2</span><span>, . . . , <em>b</em> </span><span>+ </span><em>e</em><span>}</span><span>. In this paper, we proved that several classes of regular trees, such as regular caterpillars, regular firecrackers, regular caterpillar-like trees, regular path-like trees, and regular banana trees, have a </span><em>b</em><span>-edge consecutive edge magic labeling for some </span><span>0 </span><span>< <em>b</em> < </span><span>|</span><span><em>V</em> </span><span>|</span><span>.</span></p></div></div></div>