| Summary: | <p>For a simple graph <span class="math"><em>G</em> = (<em>V</em>, <em>E</em>)</span> this paper deals with the existence of an edge labeling <span class="math"><em>φ</em> : <em>E</em>(<em>G</em>) → {0, 1, …, <em>k</em> − 1}</span>, <span class="math">2 ≤ <em>k</em> ≤ ∣<em>E</em>(<em>G</em>)∣</span>, which induces a vertex labeling <span class="math"><em>φ</em><sup> * </sup> : <em>V</em>(<em>G</em>) → {0, 1, …, <em>k</em> − 1}</span> in such a way that for each vertex <span class="math"><em>v</em></span>, assigns the label <span class="math">$\varphi(e_1)\cdot\varphi(e_2)\cdot\ldots\cdot \varphi(e_n) \pmod k$</span>, where <span class="math"><em>e</em><sub>1</sub>, <em>e</em><sub>2</sub>, …, <em>e</em><sub><em>n</em></sub></span> are the edges incident to the vertex <span class="math"><em>v</em></span>. The labeling <span class="math"><em>φ</em></span> is called a <span class="math"><em>k</em></span>-total edge product cordial labeling of <span class="math"><em>G</em></span> if <span class="math">∣(<em>e</em><sub><em>φ</em></sub>(<em>i</em>) + <em>v</em><sub><em>φ</em><sup> * </sup></sub>(<em>i</em>)) − (<em>e</em><sub><em>φ</em></sub>(<em>j</em>) + <em>v</em><sub><em>φ</em><sup> * </sup></sub>(<em>j</em>))∣ ≤ 1</span> for every <span class="math"><em>i</em>, <em>j</em></span>, <span class="math">$0 \le i &lt; j \le k-1$</span>, where <span class="math"><em>e</em><sub><em>φ</em></sub>(<em>i</em>)</span> and <span class="math"><em>v</em><sub><em>φ</em><sup> * </sup></sub>(<em>i</em>)</span> is the number of edges and vertices with <span class="math"><em>φ</em>(<em>e</em>) = <em>i</em></span> and <span class="math"><em>φ</em><sup> * </sup>(<em>v</em>) = <em>i</em></span>, respectively. The paper examines the existence of such labelings for toroidal fullerenes and for Klein-bottle fullerenes.</p>
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