模N整数环的压缩零因子图的盒维数
The boxicity of the compressed zero divisor graph of the ring of integers modulo N
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中文总结 AI 辅助
该研究确定了模N整数环的压缩零因子图的精确盒维数，解答了相关开放问题，给出了盒维数为a-1、0或a的不同条件。
中文摘要 AI 辅助
图G的盒维数，记为box(G)，是使得G为ℝ^d中轴平行盒的交图的最小整数d≥0。零因子图类由Beck于1988年提出，是一类热门图，已被多位研究者广泛研究。设Z(R)为环R的零因子集合，环R的零因子图Γ(R)定义为顶点集V(Γ(R))=Z(R)，边集E(Γ(R))={{x,y}:x,y∈Z(R)且x≠y、xy=0}。可在Γ(R)的顶点集上定义等价关系~：对顶点x和y，x~y当且仅当二者有相同零化子，即Ann(x)=Ann(y)。环R的压缩零因子图Γ_E(R)是从Γ(R)中，每个等价类仅保留一个顶点得到的简单图。本文完全回答了《离散应用数学》第391卷（2026年）第127-136页提出的两个开放问题。设正整数N的素因子分解为N=∏_{i=1}^a p_i^{n_i}，ℤ_N为模N整数环，我们确定了压缩零因子图Γ_E(ℤ_N)的精确盒维数。我们证明：当a≥2时，box(Γ_E(ℤ_N))=a-1当且仅当满足以下任一条件：(i) a≥2且N是两个互素整数x、y的乘积，其中x为无平方因子整数、y为素数的立方；(ii) a≥3且N为无平方因子整数；(iii) a≥2、N无立方因子、非无平方因子，且至少包含一个素因子p_i使得n_i=1。若a=2且n₁=n₂=1，则Γ_E(ℤ_N)是完全图，故box(Γ_E(ℤ_N))=0；在所有其他情况下，box(Γ_E(ℤ_N))=a。
英文摘要
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, $box(Γ_{E}(\mathbb{Z}_N))=a$.
