Edge magic total labeling of lexicographic product C4(2r+1) o ~K2 cycle with chords, unions of paths, and unions of cycles and paths

Abstract

<p>An <em>edge magic total (EMT) labeling</em> of a graph <span class="math"><em>G</em> = (<em>V</em>, <em>E</em>)</span> is a bijection from the set of vertices and edges to a set of numbers defined by <span class="math"><em>λ</em> : <em>V</em> ∪ <em>E</em> → {1, 2, ..., ∣<em>V</em>∣ + ∣<em>E</em>∣}</span> with the property that for every <span class="math"><em>x</em><em>y</em> ∈ <em>E</em></span>, the weight of <span class="math"><em>x</em><em>y</em></span> equals to a constant <span class="math"><em>k</em></span>, that is, <span class="math"><em>λ</em>(<em>x</em>) + <em>λ</em>(<em>y</em>) + <em>λ</em>(<em>x</em><em>y</em>) = <em>k</em></span> for some integer <span class="math"><em>k</em></span>. In this paper given the construction of an EMT labeling for certain lexicographic product <span class="math">$C_{4(2r+1)}\circ \overline{K_2}$</span>, cycle with chords <span class="math"><em></em><sup>[<em>c</em>]<em>t</em></sup><em>C</em><sub><em>n</em></sub></span>, unions of paths <span class="math"><em>m</em><em>P</em><sub><em>n</em></sub></span>, and unions of cycles and paths <span class="math"> <em>m</em>(<em>C</em><sub><em>n</em><sub>1</sub>(2<em>r</em> + 1)</sub> ∪ (2<em>r</em> + 1)<em>P</em><sub><em>n</em><sub>2</sub></sub>)</span>.</p>