Problem
Research questions and friction points this paper is trying to address.
Boxicity
Threshold Dimension
Zero Divisor Graphs
Finite Commutative Rings
Boxicity and Threshold Dimension of Zero Divisor Graphs
🤖 AI Summary
本文通过引入一种新的组合工具——整数覆盖图，解决了两类有限交换环（简约环和主理想整环的商）零因子图的盒维数和阈值维度问题。
📝 Abstract
The zero divisor graph $Γ(R)$ of a finite commutative ring $R$ has as vertices the non-zero zero divisors of $R$, with an edge between two elements exactly when their product is zero.
We determine the boxicity and threshold dimension of $Γ(R)$ for two classes of finite commutative rings: reduced rings and quotients of principal ideal domains.
Our proofs use a new combinatorial gadget, the integral covering graph, that captures the structure shared by both ring families and generalizes the disjointness graph on subsets of $[n]$, where two subsets are adjacent if and only if they are disjoint.
In doing so, we answer two questions recently posed by L.~Sunil Chandran and Suraj Kumar Sahoo in Boxicity of Zero Divisor Graphs, Discrete Applied Mathematics 391 (2026).
Innovation
Methods, ideas, or system contributions that make the work stand out.
integral covering graph
boxicity
threshold dimension
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