Boxicity of Zero Divisor Graphs

TL;DR

The paper investigates boxicity of zero-divisor graphs, focusing on Γ(ℤ_N) and Γ(R) for reduced rings. It derives exact formulae for box(Γ(ℤ_N)) in terms of a, the number of distinct primes dividing N, and shows a sharp bound dim_TH(Γ(ℤ_N)) between a−1 and a; for reduced rings it establishes ⌊k/2⌋ ≤ box(Γ(R)) ≤ dim_TH(Γ(R)) ≤ k, correcting previous lower-bound reasoning. These results rely on Roberts' graphs, threshold-interval representations, and a reduction Γ_E(R) ≅ Γ(ℤ_2^k), advancing understanding of geometric representations of algebraic graphs. The findings also raise open questions on cubicity bounds and tightness gaps between boxicity and threshold dimension, with potential implications for related graph-structure representations in algebraic contexts.

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 $Γ(R)$ for a ring $R$ is defined as the graph with the vertex set $V(Γ(R))=Z(R)$ and $E(Γ(R))=\{\{a_i,a_j\}:a_ia_j\in Z(R)\text{ and }a_ia_j=0 \}$. Let $N=Π_{i=1}^ap_i^{n_i}$ be the prime factorization of $N$. In Discrete Applied Mathematics 365 (2025), pp. 260-269, it was shown that $box(Γ(\mathbb{Z}_N))\leqΠ_{i=1}^a(n_i+1)-Π_{i=1}^a(\lfloor n_i/2\rfloor+1)-1$. In this paper we exactly determine the boxicity of $Γ(\mathbb{Z}_N)$: We show that when $N\equiv 2\pmod 4$ and $N$ is not divisible by $p^3$ for any prime divisor $p$, we have $box(Γ(\mathbb{Z}_N))=a-1$. Otherwise $box(Γ(\mathbb{Z}_N))=a$. 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 $box(Γ(R))\leq 2^k-2$. We improve this result by showing $\lfloor k/2\rfloor\leq box(Γ(R))\leq k$ with the same assumption on $R$. In this paper we also show that $a-1\leq\dim_{TH}(Γ(\mathbb{Z}_N))\leq a$ and $\lfloor k/2\rfloor\leq\dim_{TH}(Γ(R))\leq k$, where $\dim_{TH}$ is another dimensional parameter associated with graphs known as the threshold dimension.

Figures (1)

• Figure 1: A schematic diagram of the graphs $I_i$ for $i\in A$(top) and $i\in B$(bottom). Each node represents a group of vertices $F_i(j,k)$. Edges are present between two nodes if and only if every vertex in one node is adjacent to every vertex in the other node. The hollow nodes indicate that the corresponding group of vertices form an independent set and solid nodes indicate cliques. The edges inside the cliques and the edges going between two nodes account for all the edges of the graph $I_i$. On the right, the corresponding intervals are illustrated for each vertex. The intervals corresponding to vertices of hollow nodes are drawn as small disjoint intervals with thin lines and the intervals corresponding to solid nodes are drawn long and thick. Note that vertices of a particular solid node is mapped to the same long interval(hence the use of thick lines).

Theorems & Definitions (33)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4: cub
• Definition 1.5
• Theorem 1.6: Roberts1969301
• Definition 1.7
• Definition 1.8: Robert's Graph
• Definition 1.11
• Definition 1.12