On Arithmetic Cordial Labeling of Some Graphs

TL;DR

This paper introduces a new type of graph labeling called arithmetic cordial labeling modulo ta, exploring its application to various graph classes under specific conditions on associated functions.

Contribution

It defines and investigates arithmetic cordial labelings for different graphs, extending the concept with conditions on functions ta, ta, and ta.

Findings

Established conditions for arithmetic cordial labeling on star graphs.

Extended the labeling concept to ladder, cycle snake, join, corona, and tensor product graphs.

Provided criteria for the existence of such labelings under various function constraints.

Abstract

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.