Coloring of cozero-divisor graphs of commutative von Neumann regular rings

TL;DR

This paper investigates the coloring properties of cozero-divisor graphs of commutative von Neumann regular rings, establishing their perfectness and providing a formula for the clique number.

Contribution

It proves that these graphs are perfect and derives an explicit formula for their clique number, extending understanding of their combinatorial structure.

Findings

Cozero-divisor graphs of von Neumann regular rings are perfect.

An explicit formula for the clique number is provided.

The graphs are shown to be weakly perfect and perfect.

Abstract

Let $R$ be a commutative ring with non-zero identity. The cozero-divisor graph of $R$, denoted by $\Gamma^{\prime}(R)$, is a graph with vertices in $W^*(R)$, which is the set of all non-zero and non-unit elements of $R$, and two distinct vertices $a$ and $b$ in $W^*(R)$ are adjacent if and only if $a\not\in Rb$ and $b\not\in Ra$. In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given.