The book thickness of nilpotent graphs (Q2136363)

Let \(R\) be a commutative ring with nonzero identity and let \(Z_N(R)=\{y \in R: yz ~\text{is}~ \text{nilpotent for some}~z\in R^*=R \setminus \{0\} \}.\) The nilpotent graph \(\Gamma_N (R)\) of \(R\) is the graph whose vertex set coincide with \(Z_N(R)^{*}=Z_N(R)\setminus \{0\}\) and there is an edge between two distinct vertices \(x\) and \(y\) if and only if \(xy\) is nilpotent in \(R.\) An \(n\)-book is a joining of \(n\) pages (i.e., \(n\) half planes) along with the spine (i.e., a line segment). In an \(n\)-book, the embedding of a graph is to embed all the vertices linearly on a spine and every edge is embedded on any one page without crossings of the edges. The minimum number \(n\) such that the graph can be embedded in an \(n\)-book is called book thickness and it is denoted by \(bt(G).\) In this paper, the authors determine the exact value of the book thickness of all nilpotent graphs associated to commutative rings whose genus is at most one. Also, they investigate the book thickness of nilpotent graphs from local rings and some product rings.