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- W4293501550 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 $x+yin Z(R)$. 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 $mathbb{Z}_3times mathbb{Z}_3$, 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." @default.
- W4293501550 created "2022-08-29" @default.
- W4293501550 creator A5009842098 @default.
- W4293501550 creator A5060541368 @default.
- W4293501550 date "2013-05-18" @default.
- W4293501550 modified "2023-09-27" @default.
- W4293501550 title "Application of some combinatorial arrays in coloring of total graph of a commutative ring" @default.
- W4293501550 doi "https://doi.org/10.48550/arxiv.1305.4315" @default.
- W4293501550 hasPublicationYear "2013" @default.
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