On the Star Arboricity of the Zero-Divisor Graph \(\Gamma(\mathbb{Z}_{p^n})\)

A star forest is a forest each of whose components is a star. The star arboricity of a graph \(G\), denoted by \(\textrm{st}(G)\), is the minimum number of star forests whose union covers all the edges of \(G\). A nonzero element of a commutative ring \(R\) with unity is said to be a \emph{zero-divisor} of \(R\) if there exists a nonzero element \(y \in R\) such that \(xy = 0\). Given a ring \(R\) with unity, the \emph{zero-divisor graph} of \(R\), denoted by \(\Gamma(R)\), is the graph whose vertex set consists of the zero divisors of \(R\) and two vertices \(x, y \in V(\Gamma(R))\) are adjacent if and only if \(xy = 0\) in \(R\). This paper investigates the star arboricities of the zero divisor graphs \(\Gamma(\mathbb{Z}_{p^n})\), where \(n, p \in \mathbb{N}\) and \(p\) is a prime. In particular, we give bounds for \(\textrm{st}(\Gamma(\mathbb{Z}_{p^n}))\) when \(n\) is odd and determine the values of \(\textrm{st}(\Gamma(\mathbb{Z}_{p^n}))\) when \(n\) is even.