Half-Zumkeller Labeling for Some Cartesian product Graphs | ||
| Frontiers in Scientific Research and Technology | ||
| Volume 4, Issue 1, November 2022 PDF (1.46 M) | ||
| Document Type: Original Article | ||
| DOI: 10.21608/fsrt.2022.169109.1072 | ||
| Authors | ||
| Mohamed Ramadan Zeen El Deen* 1; Ghada Elmahdy2; Entesar Mohamed Elkholy3; Hamed El-Sherbiny4 | ||
| 1Department of Mathematics, Faculty of Science, Suez University, Suez, Egypt | ||
| 2Department of Basic science, Canal High Institute of Engineering and Technology, Suez 42524, Egypt | ||
| 3Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt. | ||
| 4Mathematics and Computer Science Department, Faculty of Science, Suez University | ||
| Abstract | ||
| A positive integer x is said to be a half-Zumkeller number if the proper positive divisors of x can be partitioned into two disjoint non-empty subsets of equal sum. Half-Zumkeller labeling of a graph Γ=(V(Γ),E(Γ)) with α=∣V(Γ)∣ vertices and β=∣E(Γ)∣ edges, is an injective mapping ψ of the vertex set V(Γ) into the set of natural number such that the induced mapping 〖 ψ〗^*:E(Γ)⟶Z^+∪{0}, given by ψ^* (λ μ)=ψ (λ) ψ ( μ), is a half-Zumkeller number for all λ μ∈E(Γ), λ,μ∈v(Γ) . The graph that admits a half-Zumkeller labeling is called a half-Zumkeller graph. In this paper, we present half-Zumkeller labeling of the graphs: the stacked book graph SB_(m,n) , the cylinder grid graph C_(m,n) and the prisms of the following graphs: ladder graph L_n , the grid graph G_(m,n) , the gear graph G_n , flower graph FL_n . Furthermore, if H a non-totally disconnected subgraph of Γ then H is also, a half- Zumkeller graph. | ||
| Keywords | ||
| Zumkeller labeling; half-Zumkeller labeling; Stacked book graph; the cylinder grid graph C_(m,n) | ||
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