On Arithmetic Cordial Labeling of Some Graphs

Abstract

Let $\hat{I}\cdot$ 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 $\hat{I}{\P }_{\hat{I}}\cdot :\{\hat{I}¾\in \mathbb{Z}:\gcd (\hat{I}¾,\hat{I}\cdot )=1\}\to \{0,1\}$ 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 $\hat{I}\cdot$ under $⟨S,\hat{I}{\P }_{\hat{I}}\cdot ,\star ⟩$ if the induced function $f_{\hat{I}}{\cdot }^{\ast }:E(G)\to \{0,1\}$, defined by $f_{\hat{I}}{\cdot }^{\ast }(uv)=0$ whenever $\hat{I}{\P }_{\hat{I}}\cdot (f(a)\star f(b))=0$ or $\gcd (f(a)\star f(b),\hat{I}\cdot )\ne 1$, and $f_{\hat{I}}{\cdot }^{\ast }(uv)=1$ whenever $\hat{I}{\P }_{\hat{I}}\cdot (f(a)\star f(b))=1$, satisfies the condition $|e_{f_{\hat{I}}{\cdot }^{\ast }}(0)-e_{f_{\hat{I}}{\cdot }^{\ast }}(1)|\le 1$, where $e_{f_{\hat{I}}{\cdot }^{\ast }}(i)$ 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 $\hat{I}{\P }_{\hat{I}}\cdot$. The graphs included are star graphs, ladder graphs, alternate cycle snake graphs, join graphs, corona graphs, and tensor product graphs.

19 pages