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 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, 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.
