On Arithmetic Cordial Labeling of Some 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 connected 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 $⟨S,{\zeta }_{\eta },\star ⟩$ if the induced function $f_{\eta }^{\ast }:E(G)\to \{0,1\}$, defined by $f_{\eta }^{\ast }(uv)=0$ whenever ${\zeta }_{\eta }(f(a)\star f(b))=0$ or $\gcd (f(a)\star f(b),\eta )\mathrm{\ne }1$, and $f_{\eta }^{\ast }(uv)=1$ whenever ${\zeta }_{\eta }(f(a)\star f(b))=1$, 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, we explore the arithmetic cordial labeling of some graphs under conditions imposed on the function ${\zeta }_{\eta }$. The graphs included are star graphs, ladder graphs, alternate cycle snake graphs, join graphs, corona graphs, and tensor product graphs.