Rings whose total graphs have small vertex-arboricity and arboricity

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