On Arithmetic Cordial Labeling of Product Graphs

Abstract

Let $\eta$ 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 $\zeta_{\eta}:\{\xi\in \mathbb{Z}:\gcd(\xi,\eta)=1\}\to \{0,1\}$ 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 $\eta$ under the arithmetic structure $\langle S,\zeta_\eta,\star\rangle$ if the induced function $f_\eta^*:E(G)\to \{0,1\}$, defined by $f_\eta^*(ab)=1$ whenever $\gcd(f(a)\star f(b),\eta)= 1$ and $\zeta_\eta(f(a)\star f(b))=1$; otherwise, $f_\eta^*(ab)=0$, satisfies the condition $|e_{f_\eta^*}(0)-e_{f_\eta^*}(1)|\leq 1$, where $e_{f_\eta^*}(i)$ 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.