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

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.