k-Zumkeller labeling of the Cartesian and tensor product of paths and cycles

Article type: Research Article

Authors: Basher, M.^{; *}

Affiliations: Department of Mathematics and Computer Science, Faculty of Science, Suez University, Suez, Egypt

Correspondence: [*] Corresponding author. M. Basher, Department of Mathematics and Computer Science, Faculty of Science, Suez University, Suez, Egypt. E-mail: [email protected].

Abstract: A k-Zumkeller labeling for the graph G = (V, E) is an assignment f of a label to each vertices of G such that each edge uv ∈ E is assigned the label f (u) f (v), the resulting edge labels are k distinct Zumkeller numbers. In this paper, we prove that the graph Pm × Pn is k-Zumkeller graph for m, n ≥ 3 while Pm × Cn and Cm × Cn are k-Zumkeller graphs for n ≡ 4 (mod2). Also we show that the graphs Pm ⊗ Pn and Pm ⊗ Cn for m, n ≥ 3 admit k-Zumkeller labeling. Further, the graph Cm ⊗ Cn where m or n is even admit a k-Zumkeller labeling.

Keywords: Zumkeller number, k-Zumkeller labeling, Cartesian and tensor product of graphs

Keywords: 05C78

DOI: 10.3233/JIFS-201765

Journal: Journal of Intelligent & Fuzzy Systems, vol. 40, no. 3, pp. 5061-5070, 2021