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On Arithmetic Cordial Labeling of Some Graphs

Jason D. Andoyo, Jemina Clarisse C. Prudencio, Ricky F. Rulete

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.

On Arithmetic Cordial Labeling of Some Graphs

Abstract

Let be a fixed positive integer. Let be a subset of , be a binary function, and be a function. For a simple connected graph of order , a bijective function (where ) is called an arithmetic cordial labeling modulo under if the induced function , defined by whenever or , and whenever , satisfies the condition , where is the number of edges with label (). In this paper, we explore the arithmetic cordial labeling of some graphs under conditions imposed on the function . The graphs included are star graphs, ladder graphs, alternate cycle snake graphs, join graphs, corona graphs, and tensor product graphs.
Paper Structure (5 sections, 14 theorems, 90 equations)

This paper contains 5 sections, 14 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Basic Concepts
  3. General Results
  4. Arithmetic Cordial Labeling modulo $p$ under $\left\langle S,\zeta_p,+\right\rangle$
  5. Arithmetic Cordial Labeling modulo $p$ under $\left\langle S,\zeta_p,\cdot\right\rangle$

Key Result

Theorem 3.1

Let $G$ be a simple connected graph and let $\eta_1$ and $\eta_2$ be fixed integers. Assume that $\langle S_1,\zeta_{\eta_1}^1,\star_1\rangle\cong \langle S_2,\zeta_{\eta_2}^2,\star_2\rangle$. Then $G$ is an arithmetic cordial graph modulo $\eta_1$ under $\langle S_1,\zeta_{\eta_1}^1,\star_1\rangle$

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 3.1
  • ...and 28 more