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 ψ: E(G) → {0, 1, . . ., k-1}, 2 ≤ k ≤ |E(G)|, which induces a vertex labeling ψ*: V (G) → {0; 1;: :: ; k-1} in such a way that for each vertex v, assigns the label ψ(e1) · ψ(e2) ·: :: · ψ(en) (mod k), where e1, e2, . . ., en are the edges incident to the vertex v. The labeling ψ is called a k-total edge product cordial labeling of G if |(eψ(i) + vψ*(i)) - (eψ(j) + vψ*(j))| ≤ 1 for every i; j, 0 ≤ i < j ≤ k - 1, where eψ(i) and vψ*(i) is the number of edges and vertices with ψ(e) = i and ψ*(v) = i, respectively. The paper examines the existence of such labelings for toroidal fullerenes and for Klein-bottle fullerenes.