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 $d\geq 0$ 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 $Γ(R)$ for a ring $R $ is defined as the graph with the vertex set $V(Γ(R))=Z(R)$ and $E(Γ(R))=\{\{x,y\}\colon x,y\in Z(R)\text{ with }x\neq y\text{ and }x y=0\}$. One can define an equivalence relation $\sim$ on $V(Γ(R))$ 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 $Γ_E(R)$ for a ring $R$ is the simple graph obtained from $Γ(R)$ 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 $Γ_E(\mathbb{Z}_N)$. We show that when $a\geq 2$, $box(Γ_E(\mathbb{Z}_N))= a-1$ if and only if one of the following is true: $(i)$ $a\geq 2$ 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; $(ii)$ $a\geq 3$ and $N$ is square-free; $(iii)$ $a\geq 2$, $N$ 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 $Γ_{E}(\mathbb{Z}_N)$ is a clique, and so, $box(Γ_{E}(\mathbb{Z}_N))=0$. In all other cases, $box(Γ_{E}(\mathbb{Z}_N))=a$.