No abstract

<p>Let <span class="math"><em>H</em></span> and <span class="math"><em>G</em></span> be two simple graphs. The concept of an <span class="math"><em>H</em></span>-magic decomposition of <span class="math"><em>G</em></span> arises from the combination between graph decomposition and graph labeling. A decomposition of a graph <span class="math"><em>G</em></span> into isomorphic copies of a graph <span class="math"><em>H</em></span> is <span class="math"><em>H</em></span>-magic if there is a bijection <span class="math"><em>f</em> : <em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>) → {1, 2, ..., ∣<em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>)∣}</span> such that the sum of labels of edges and vertices of each copy of <span class="math"><em>H</em></span> in the decomposition is constant. A lexicographic product of two graphs <span class="math"><em>G</em>₁</span> and <span class="math"><em>G</em>₂, </span> denoted by <span class="math"><em>G</em>₁[<em>G</em>₂], </span> is a graph which arises from <span class="math"><em>G</em>₁</span> by replacing each vertex of <span class="math"><em>G</em>₁</span> by a copy of the <span class="math"><em>G</em>₂</span> and each edge of <span class="math"><em>G</em>₁</span> by all edges of the complete bipartite graph <span class="math"><em>K</em>_{<em>n</em>, <em>n</em>}</span> where <span class="math"><em>n</em></span> is the order of <span class="math"><em>G</em>₂.</span> In this paper we provide a sufficient condition for <span class="math">$\overline{C_{n}}[\overline{K_{m}}]$</span> in order to have a <span class="math">$P_{t}[\overline{K_{m}}]$</span>-magic decompositions, where <span class="math"><em>n</em> &gt; 3, <em>m</em> &gt; 1, </span> and <span class="math"><em>t</em> = 3, 4, <em>n</em> − 2</span>.</p>