Boxicity of Zero Divisor Graphs

Abstract

A $d$-dimensional box is the cartesian product $R_i\times\cdots\times R_d$ where each $R_i$ is a closed interval on the real line. The boxicity of a graph, denoted as $box(G)$, is the minimum integer $d\geq 0$ such that $G$ is the intersection graph of a collection of $d$-dimensional boxes. The study of graph classes associated with algebraic structures is a fascinating area where graph theory and algebra meet. A well-known class of graphs associated with rings is the class of zero divisor graphs introduced by Beck in 1988. Since then, this graph class has been studied extensively by several researchers. Denote by $Z(R)$ 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 $E(\Gamma(R))=\{\{a_i,a_j\}:a_ia_j\in Z(R)\text{ and }a_ia_j=0 \}$. Let $N=\Pi_{i=1}^ap_i^{n_i}$ be the prime factorization of $N$. In Discrete Applied Mathematics 365 (2025), pp. 260-269, it was shown that $box(\Gamma(\mathbb{Z}_N))\leq\Pi_{i=1}^a(n_i+1)-\Pi_{i=1}^a(\lfloor n_i/2\rfloor+1)-1$. In this paper we exactly determine the boxicity of $\Gamma(\mathbb{Z}_N)$: We show that when $N\equiv 2\pmod 4$ and $N$ is not divisible by $p^3$ for any prime divisor $p$, we have $box(\Gamma(\mathbb{Z}_N))=a-1$. Otherwise $box(\Gamma(\mathbb{Z}_N))=a$. Suppose $R$ is a non-zero commutative ring with identity that is also a reduced ring and let $k$ be the size of the set of minimal prime ideals of $R$. In the same paper, it was showed that $box(\Gamma(R))\leq 2^k-2$. We improve this result by showing $\lfloor k/2\rfloor\leq box(\Gamma(R))\leq k$ with the same assumption on $R$. In this paper we also show that $a-1\leq\dim_{TH}(\Gamma(\mathbb{Z}_N))\leq a$ and $\lfloor k/2\rfloor\leq\dim_{TH}(\Gamma(R))\leq k$, where $\dim_{TH}$ is another dimensional parameter associated with graphs known as the threshold dimension.