Vertex and region colorings of planar idempotent divisor graphs of commutative rings.

Nazar H Shuker Nazar H Shuker,Mohammed Authman Mohammed Authman,Husam Q Mohammad Husam Q Mohammad

doi:10.52866/ijcsm.2022.01.01.008

Nazar H Shuker Nazar H Shuker, Mohammed Authman Mohammed Authman + Show 1 more

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https://doi.org/10.52866/ijcsm.2022.01.01.008

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Abstract

The idempotent divisor graph of a commutative ring R is a graph with vertices set in R* = R-{0}, and any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e2 = e ϵ R, and is denoted by Л(R). The purpose of this work is using some properties of ring theory and graph theory to find the clique number, the chromatic number and the region chromatic number for every planar idempotent divisor graphs of commutative rings. Also we show the clique number is equal to the chromatic number for any planar idempotent divisor graph. Among other results we prove that: Let Fq, Fpa are fieldes of orders q and pa respectively, where q=2 or 3, p is a prime number and a Is a positive integer. If a ring R @ Fq x Fpa . Then (Л(R))= (Л(R)) = *( Л(R)) = 3.

Highlights

In this work, R is a finite commutative ring with an identity of 1= 0, and Z(R) is a set of all-zero divisors
The idempotent divisor graph of a commutative ring R is a graph with vertices set in R * = R-{0}, and any distinct vertices x and y are adjacent if and only if x.y = e
The purpose of this work is to use some properties of ring theory and graph theory to determine the clique number, the chromatic number and the region chromatic number for each planar idempotent divisor graph of the commutative rings. we show that the clique number is equal to the chromatic number for any planar idempotent divisor graph

Summary

INTRODUCTION

R is a finite commutative ring with an identity of 1= 0, and Z(R) is a set of all-zero divisors. In 2021, Mohammad and Shuker [10] defined the idempotent divisor graph of commutative ring R with identity 1= 0. This graph is denoted by Π(R), which has vertices set in R* = R-{0}. We use ring theory and graph theory to determine the clique number, the chromatic number and the region chromatic number for each planar idempotent divisor graph of commutative rings. We elaborate the region chromatic numbers for planar idempotent divisor graphs of the commutative rings. Of ω(Π(R)), χ(Π(R)) and χ (Π(R)) for any commutative ring that has a planar idempotent divisor graph

VERTEX COLOURING

REGION CHROMATIC NUMBERS FOR PLANAR IDEMPOTENT DIVISOR GRAPHS OF LOCAL RINGS

CONCLUSION