Rings whose total graphs have small vertex-arboricity and arboricity

Year 2021, Volume: 50 Issue: 1, 110 - 119, 04.02.2021

Morteza Fatehi Kazem Khashyarmanesh Abbas Mohammadian

https://doi.org/10.15672/hujms.589753

Abstract

Let $R$ be a commutative ring with non-zero identity, and $Z(R)$ be its set of all zero-divisors. The total graph of $R$, denoted by $T(\Gamma(R))$, is an undirected graph with all elements of $R$ as vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. In this article, we characterize, up to isomorphism, all of finite commutative rings whose total graphs have vertex-arboricity (arboricity) two or three. Also, we show that, for a positive integer $v$, the number of finite rings whose total graphs have vertex-arboricity (arboricity) $v$ is finite.

Keywords

Total graph‎, ‎arboricity‎, vertex-arboricity

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