On Arithmetic Cordial Labeling of Product Graphs

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 $\mathrm{\mid }S\mathrm{\mid }=n$) is called an arithmetic cordial labeling modulo $\eta$ under the arithmetic structure $⟨S,{\zeta }_{\eta },\star ⟩$ if the induced function $f_{\eta }^{\ast }:E(G)\to \{0,1\}$, defined by $f_{\eta }^{\ast }(ab)=1$ whenever $\gcd (f(a)\star f(b),\eta )=1$ and ${\zeta }_{\eta }(f(a)\star f(b))=1$; otherwise, $f_{\eta }^{\ast }(ab)=0$, satisfies the condition $\mathrm{\mid }{e}_{f_{\eta }^{\ast }}(0)-e_{f_{\eta }^{\ast }}(1)\mathrm{\mid }\le 1$, where $e_{f_{\eta }^{\ast }}(i)$ 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.