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 $|S|=n$) is called an arithmetic cordial labeling modulo $\eta$ under $\langle S,\zeta_\eta,\star\rangle$ if the induced function $f_\eta^*:E(G)\to \{0,1\}$, defined by $f_\eta^*(uv)=0$ whenever $\zeta_\eta(f(a)\star f(b))=0$ or $\gcd(f(a)\star f(b),\eta)\neq 1$, and $f_\eta^*(uv)=1$ whenever $\zeta_\eta(f(a)\star f(b))=1$, 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, 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.