Boxicity of Zero Divisor Graphs

The problem asks for exact values and structural descriptions of boxicity and threshold dimension for zero-divisor graphs of finite commutative rings. No proposer or original date is identified in the retrieved sources.

Known results

• For $N=\prod_{i=1}^{a}p_i^{n_i}$, the 2025 preprint claims $\operatorname{box}(\Gamma(\mathbb{Z}_N))=a-1$ in the stated exceptional case and $a$ otherwise.
• It claims $a-1\leq\dim_{\mathrm{TH}}(\Gamma(\mathbb{Z}_N))\leq a$.
• For reduced rings with $k$ minimal primes, it claims $\lfloor k/2\rfloor\leq\operatorname{box}(\Gamma(R))\leq\dim_{\mathrm{TH}}(\Gamma(R))\leq k$; closing this gap remains open.
• The preprint reports that Kavaskar’s earlier lower-bound proof and one boxicity-one characterization were incorrect, citing $N=18$ as a counterexample.

August 27, 2026 integral-covering-graph development

A new arXiv paper introduces integral covering graphs, answers two structural questions for two broad specified classes of finite commutative rings, and connects the graphs with disjointness graphs. This is claimed progress rather than a verified solution of the general problem.

Current status (as of August 2026): The cyclic-ring formula and reduced-ring bounds are claimed, and a newer paper claims structural results for two broad ring classes, but exact values for general finite commutative rings and several reduced-ring questions remain open.