On total edge product cordial labeling of fullerenes

Abstract

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