On Arithmetic Cordial Labeling of Some Graphs

Abstract

Let $η$ 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 $ζ_η:\{ξ\in \mathbb{Z}:\gcd(ξ,η)=1\}\to \{0,1\}$ 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 $η$ under $\langle S,ζ_η,\star\rangle$ if the induced function $f_η^*:E(G)\to \{0,1\}$, defined by $f_η^*(uv)=0$ whenever $ζ_η(f(a)\star f(b))=0$ or $\gcd(f(a)\star f(b),η)\neq 1$, and $f_η^*(uv)=1$ whenever $ζ_η(f(a)\star f(b))=1$, satisfies the condition $|e_{f_η^*}(0)-e_{f_η^*}(1)|\leq 1$, where $e_{f_η^*}(i)$ 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 $ζ_η$. The graphs included are star graphs, ladder graphs, alternate cycle snake graphs, join graphs, corona graphs, and tensor product graphs.