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 e₁, e₂, …, 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, $0 \le i < j \le k-1$, 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.