## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 46, Number 3 (2017), 395-406.

### The extended zero-divisor graph of a commutative ring II

M. BAKHTYIARI, M. J. NIKMEHR, and R. NIKANDISH

#### Abstract

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The extended zero-divisor graph of $R$ is the undirected (simple) graph $\Gamma'(R)$ with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if either $Rx\cap \mathrm{Ann}(y)\neq (0)$ or $Ry\cap \mathrm{Ann}(x)\neq (0)$. In this paper, we continue our study of the extended zero-divisor graph of a commutative ring that was introduced in [4]. We show that the extended zero-divisor graph associated with an Artinian ring is weakly perfect, i.e., its vertex chromatic number equals its clique number. Furthermore, we classify all rings whose extended zero-divisor graphs are planar.

#### Article information

**Source**

Hokkaido Math. J., Volume 46, Number 3 (2017), 395-406.

**Dates**

First available in Project Euclid: 7 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1510045304

**Digital Object Identifier**

doi:10.14492/hokmj/1510045304

**Mathematical Reviews number (MathSciNet)**

MR3720335

**Zentralblatt MATH identifier**

06814869

**Subjects**

Primary: 13B99: None of the above, but in this section 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 05C69: Dominating sets, independent sets, cliques

**Keywords**

Extended zero-divisor graph Clique number Chromatic number Planar graph

#### Citation

BAKHTYIARI, M.; NIKMEHR, M. J.; NIKANDISH, R. The extended zero-divisor graph of a commutative ring II. Hokkaido Math. J. 46 (2017), no. 3, 395--406. doi:10.14492/hokmj/1510045304. https://projecteuclid.org/euclid.hokmj/1510045304