For every positive integer N=∏_i=1^ap_i^n_i, where the p_i are distinct primes and n_i≥ 1, determine the boxicity operatornamebox(Gamma_E(Z_N)) of the compressed zero-divisor graph. Here Gamma_E(Z_N) is obtained from the zero-divisor graph of Z_N by identifying zero divisors x and y whenever operatornameAnn(x)=operatornameAnn(y) and retaining one representative from each equivalence class.

Open questions on the boxicity of compressed zero-divisor graphs of Z_N

For every positive integer $N=\prod_{i=1}^{a}p_i^{n_i}$ , where the $p_i$ are distinct primes and $n_i\geq 1$ , determine the boxicity $\operatorname{box}(\Gamma_E(\mathbb{Z}_N))$ of the compressed zero-divisor graph. Here $\Gamma_E(\mathbb{Z}_N)$ is obtained from the zero-divisor graph of $\mathbb{Z}_N$ by identifying zero divisors $x$ and $y$ whenever $\operatorname{Ann}(x)=\operatorname{Ann}(y)$ and retaining one representative from each equivalence class.

References

Primary source

arXiv

Additional references

The boxicity of the compressed zero divisor graph of the ring of integers modulo N — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new preprint claims to settle both questions by giving a complete arithmetic classification, but the result has not yet been independently verified.

The problem asks for the boxicity of compressed zero-divisor graphs of $\mathbb{Z}_N$ and records two explicitly posed questions about this invariant.

August 24, 2026 preprint

A preprint claims a complete arithmetic classification of the boxicity of the compressed zero-divisor graph of $\mathbb{Z}_N$ , which would close both questions. The claim is unrefereed and remains unverified. Earlier material found in the scan classifies the ordinary zero-divisor graph, not the compressed graph, so it does not independently establish this result.

Current status (as of August 2026): A preprint claims the compressed-graph questions are solved, but the classification and its proof remain unverified.

Sources

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