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