Application of some combinatorial arrays in coloring of total graph of a commutative ring

TL;DR

This paper investigates the coloring properties of the total graph of a finite commutative ring, establishing conditions under which the chromatic and clique numbers are equal and explicitly determining these numbers for various subgraphs.

Contribution

It provides new results on the chromatic and clique numbers of total and induced zero-divisor graphs of finite rings under specific algebraic conditions.

Findings

Chromatic and clique numbers of the total graph equal the maximum size of maximal ideals.

Determined chromatic and clique numbers for zero-divisor and regular subgraphs.

Results depend on the characteristic of residue fields and the structure of the ring.

Abstract

Let $R$ be a commutative ring with unity and $Z(R)$ and ${\rm Reg}(R)$ be the set of zero-divisors and non-zero zero-divisors of $R$, respectively. We denote by $T(\Gamma(R))$, the total graph of $R$, a simple graph with the vertex set $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. The induced subgraphs on $Z(R)$ and ${\rm Reg}(R)$ are denoted by $Z(\Gamma(R))$ and $Reg(\Gamma(R))$, respectively. These graphs were first introduced by D.F. Anderson and A. Badawi in 2008. In this paper, we prove the following result: let $R$ be a finite ring and one of the following conditions hold: (i) The residue field of $R$ of minimum size has even characteristic, (ii) Every residue field of $R$ has odd characteristic and $\frac{R}{J(R)}$ has no summand isomorphic to $\mathbb{Z}_3\times \mathbb{Z}_3$, then the chromatic number and clique number of $T(\Gamma(R))$ are equal to $\max\{|\mathfrak{m}|\,:\, \mathfrak{m}\in {\rm Max}(R)\}$. The same result holds for $Z(\Gamma(R))$. Moreover, if the residue field of $R$ of minimum size has even characteristic or every residue field of $R$ has odd characteristic, then we determine the chromatic number and clique number of $Reg(\Gamma(R))$ as well.