On Arithmetic Cordial Labeling of Product Graphs

Abstract

Let $η$ be a fixed positive integer. Let $S$ be a subset of $\mathbb{Z}$, $\star:S\times S\to \mathbb{Z}$ be a binary function, and $ζ_η:\{ξ\in \mathbb{Z}:\gcd(ξ,η)=1\}\to \{0,1\}$ be a function. For a simple graph $G$ of order $n$, a bijective function $f:V(G)\to S$ (where $|S|=n$) is called an arithmetic cordial labeling modulo $η$ under the arithmetic structure $\langle S,ζ_η,\star\rangle$ if the induced function $f_η^*:E(G)\to \{0,1\}$, defined by $f_η^*(ab)=1$ whenever $\gcd(f(a)\star f(b),η)= 1$ and $ζ_η(f(a)\star f(b))=1$; otherwise, $f_η^*(ab)=0$, satisfies the condition $|e_{f_η^*}(0)-e_{f_η^*}(1)|\leq 1$, where $e_{f_η^*}(i)$ is the number of edges with label $i$ ($i=0,1$). In this paper, the arithmetic cordial labeling of product graphs, namely, corona, lexicographic, cartesian, tensor, and strong, is explored under the operation of addition.