The boxicity of the compressed zero divisor graph of the ring of integers modulo N

Abstract

The boxicity of a graph $G$, denoted by $box(G)$, is the minimum integer $d\ge 0$ such that $G$ is the intersection graph of axis-parallel boxes in ${\mathbb{R}}^d$. The class of zero divisor graphs introduced by Beck (1988) is a popular class of graphs and has been studied extensively by several researchers. Suppose $Z(R)$ is the set of zero divisors of a ring $R$. The zero divisor graph $\Gamma (R)$ for a ring $R$ is defined as the graph with the vertex set $V(\Gamma (R))=Z(R)$ and $E(\Gamma (R))=\{\{x,y\}:x,y\in Z(R)\text{ with }x\ne y\text{ and }xy=0\}$. One can define an equivalence relation $\sim$ on $V(\Gamma (R))$ such that for vertices $x$ and $y$, one has $x\sim y$ if and only if $x$ and $y$ have the same annihilator, i.e., $Ann(x)=Ann(y)$. The compressed zero divisor graph ${\Gamma }_E(R)$ for a ring $R$ is the simple graph obtained from $\Gamma (R)$ by retaining exactly one vertex from each equivalence class induced by $\sim$. In this paper, we completely answer two open questions posed in Discrete Applied Mathematics 391 (2026), pp. 127-136. Let $N={\prod }_{i=1}^a{p}_i^{n_i}$ be the prime factorization of a positive integer $N$ and let ${\mathbb{Z}}_N$ be the ring of integers modulo $N$. We determine the exact boxicity of the compressed zero divisor graph ${\Gamma }_E({\mathbb{Z}}_N)$. We show that when $a\ge 2$, $box({\Gamma }_E({\mathbb{Z}}_N))=a-1$ if and only if one of the following is true: $(i)$ $a\ge 2$ and $N$ is the product of two coprime integers $x$ and $y$ such that $x$ is a square-free integer and $y$ is the cube of a prime number; $(ii)$ $a\ge 3$ and $N$ is square-free; $(iii)$ $a\ge 2$, $N$ is cube-free, not square-free, and contains at least one prime divisor $p_i$ such that $n_i=1$. If $a=2$ and $n_1=n_2=1$, then ${\Gamma }_E({\mathbb{Z}}_N)$ is a clique, and so, $box({\Gamma }_E({\mathbb{Z}}_N))=0$. In all other cases, $box({\Gamma }_E({\mathbb{Z}}_N))=a$.