Boxicity and Threshold Dimension of Zero Divisor Graphs

Marco Caoduro; Meike Neuwohner

REVIEW 1 major objections 5 minor 26 references

Boxicity and Threshold Dimension of Zero Divisor Graphs

T0 review · 1 major / 5 minor · reviewed 2026-08-28 · deepseek-v4-flash

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Pith's one-line read The boxicity and threshold dimension of zero divisor graphs are n−1 or n, with the cutoff set by lonely minimal primes.

desk verdict The integral-covering-graph framework is promising, but the main theorem is false: for R=F2×F2×F2 the zero divisor graph is 3K2, an interval graph, so box=1, not the predicted 2. read the letter →

arxiv 2608.27381 v1 pith:BSHPWUO6 submitted 2026-08-27 math.CO cs.DMmath.AC

Marco Caoduro , Meike Neuwohner This is my paper · ORCID

classification math.COcs.DMmath.AC MSC 05C6205C7513A15

keywords zerodivisorgraphboxicitythresholddimensionreducedringprincipalidealdomainintegralcoveringdisjointnessminimalprime

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes the exact boxicity and threshold dimension of zero divisor graphs for two families of finite commutative rings: reduced rings and finite quotients of principal ideal domains. For a reduced ring with n minimal prime ideals, both parameters are n−1 if some minimal prime is "lonely" and n otherwise, with small-n cases listed separately, so the earlier n/2 lower bound is not tight. For PID quotients, the exact value is again n or n−1 in nearly all cases, with the cutoff expressed through the exponents m_i and the sizes of the residue fields. The proofs go through a new combinatorial graph, the integral covering graph D(m), which the zero divisor graphs of both families reduce to, and whose boxicity and threshold dimension are computed exactly. This answers two recently posed open questions.

What carries the argument

The central object is the integral covering graph $D(m)$, defined for a vector $m\in\mathbb{N}_{\ge1}^n$ as the graph whose vertices are all vectors in $\prod_{i=1}^n\{0,\dots,m_i\}$ except $0$ and $m$, with two vertices adjacent exactly when $v_i+w_i\ge m_i$ for every coordinate $i$. It specializes to the disjointness graph $D_n$ on nonempty proper subsets of $[n]$ when $m=1^n$. The machinery has three parts: coordinate encodings (membership flags in minimal prime ideals for reduced rings, exponent vectors in the PID-quotient case) under which zero product becomes a coordinatewise sum inequality; surjectivity lemmas guaranteeing that every admissible coordinate vector is realized by an actual zero divisor, which make the compressed zero divisor graphs isomorphic to $D(1^n)$ and $D(m)$; and matching combinatorial bounds, with lower bounds coming from induced n-fold joins of $K_2$ and partial-join threshold arguments, and upper bounds from explicit interval and threshold representations. The identity $\mathrm{box}(G^E)=\mathrm{box}(G^{E'})$ for reduced graphs transfers the answers back from the compressed graph to $\Gamma(R)$ or $\Gamma(Q)$.

What would settle it

The most direct check is on the graph $D(1^4)$, the disjointness graph of the 14 nonempty proper subsets of $\{1,2,3,4\}$: Theorems 42 and 10 predict its boxicity is 3, so a computer search that writes $D(1^4)$ as the edgewise intersection of two interval graphs would refute the combinatorial core, and via the identification $\Gamma(\mathbb{F}_2^4)^E\cong D(1^4)$ it would also refute the ring theorem's value for $\mathbb{F}_2^4$.

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Extended reading notes

Core claim

On the paper's own terms, the central result is Theorem 8: if R is a finite commutative reduced ring with n minimal prime ideals, then for n≥3, $$\mathrm{box}(\Gamma(R))=\dim_{\mathrm{TH}}(\Gamma(R))=\begin{cases} n-1 & \text{if some minimal prime ideal is lonely,}\\ n & \text{otherwise,}\end{cases}$$ with the n=1 and n=2 cases stated separately (0; and 0, 1, or 2 according to loneliness). For a finite quotient Q of a PID, the answer is governed by the factorization of the generator: when $g=\prod_{i=1}^n p_i^{m_i}$, Theorems 79 and 80 give $\mathrm{box}(\Gamma(Q))$ and $\dim_{\mathrm{TH}}(\Gamma(Q))$ as n, except in small-exponent, small-residue-field cases where the value drops to n−1, 1, or 0. Both ring theorems are consequences of a single combinatorial theorem for integral covering graphs: for $m\in\mathbb{N}_{\ge1}^n$, the graph $D(m)$ has boxicity and threshold dimension given exactly by m (Theorems 42 and 43), and the disjointness graph on the nonempty proper subsets of $[n]$ is the special case $D(1^n)$. The proof route is: encode ring elements by coordinate vectors, prove every admissible vector is realized, identify the compressed zero divisor graph with $D(1^n)$ or $D(m)$, compute the combinatorial parameter, and transfer it back.

Load-bearing premise

The load-bearing premise is that the rings are full in a combinatorial sense: every nonempty proper subset of the minimal prime ideals occurs as the exact set of minimal primes containing some zero divisor, and in PID quotients every exponent vector below m occurs; if one pattern were missing, the lower-bound constructions would lose a vertex and the n or n−1 formulas could fail.

Editorial extensions

If this is right

For a reduced ring with n minimal primes, the earlier lower bound n/2 is replaced by the exact value n−1 or n; in particular, that lower bound is not tight for n≥3.
For finite quotients of PIDs, both $\mathrm{box}(\Gamma(Q))$ and $\dim_{\mathrm{TH}}(\Gamma(Q))$ are computed by a finite arithmetic condition on the exponents $m_i$ and residue field sizes $|R/p_iR|$, with no search over representations needed.
The zero divisor graph of $\mathbb{Z}/M\mathbb{Z}$, including its compressed graph, now has exact boxicity and threshold dimension for every M, covering the previous partial results and answering the open question about compressed graphs.
One combinatorial theorem for $D(m)$ drives both ring families, so results such as $\mathrm{box}(D(1^n))=n-1$ for n≥3 transfer directly to reduced rings and to the squarefree case of PID quotients.
There are integral covering graphs whose boxicity and threshold dimension differ by exactly one, for example $D(2,1)$, so the two parameters are genuinely different at the combinatorial level.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not stated in the paper, but following from the finite reduced ring decomposition into fields, the lonely condition for a minimal prime ideal $M_i$ should be equivalent to $|R/M_i|=2$; under that equivalence, the reduced-ring formula and the PID-quotient formula are the same $\mathbb{F}_2$ cutoff, with binary residue fields being the only ones that can reduce the dimension.
The integral covering graphs offer a natural test family for the gap between boxicity and threshold dimension: since the paper shows the gap can be exactly one (e.g. $D(2,1)$), these graphs could serve as counterexamples to any conjecture that the two parameters coincide for graph classes built from coordinatewise inequalities.
Because the PID theorem applies to any PID, the same n/n−1 numerology should hold for finite quotients of rings such as $\mathbb{F}_q[x]$; a testable extension is whether the formulas persist for finite quotients of one-dimensional domains that are not PIDs, where the surjectivity of the exponent map is exactly what can fail.

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Formalized claims in Lean

Claim #1: On the paper's own terms, the central result is Theorem 8: if R is a finite commutative reduced ring with n minimal prime ideals, then for n≥3, $$\mathrm{box}(\Gamma(R))=\dim_{\mathrm{TH}}(\Gamma(R))=\begin{cases} n-1 & \text{if some minimal prime ideal is lonely,}\\ n & \text{otherwise,}\end{cases}$$ with the n=1 and n=2 cases stated separately (0; and 0, 1, or 2 according to loneliness). For a f

/-- @claim 1 On the paper's own terms, the central result is Theorem 8: if R is a finite commutative reduced ring with n minimal prime ideals, then for n≥3, $$\mathrm{box}(\Gamma(R))=\dim_{\mathrm{TH}}(\Gamma(R))=\begin{cases} n-1 & \text{if some minimal prime ideal is lonely,}\\ n & \text{otherwise,}\end{cases}$$ with the n=1 and n=2 cases stated separately (0; and 0, 1, or 2 according to loneliness). For a f -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things. First, the paper has a real idea: the integral covering graph D(m) is a sensible common generalization of the disjointness graph and the coordinate vectors behind zero divisor graphs of reduced rings and PID quotients. The upper-bound constructions in Section 3.2 are careful and reusable, and the ring-to-combinatorics reductions (Lemma 56, Proposition 71) look right. Second, the main theorem is false as stated. The stress-test note is correct: D(1^3) is exactly 3K2, which is an interval graph, so box(D(1^3))=1. For R=F2×F2×F2, the zero divisor graph is D(1^3), all three minimal prime ideals are lonely, and Theorem 8 predicts box=2. So n=3 kills the paper.

The reader's report caught a symptom of this: Lemma 46 assumes every handled pair is left- or right-handled, but I_{a,b} may sit between I_a and I_b. The bigger problem is Lemma 41 and Theorem 10, which assert box(D_n)=n−1 for n≥3. That is already false at n=3. Since the lower bounds for reduced rings and PID quotients are built on that lemma, the n≥3 case of Theorem 8 and the corresponding parts of Theorems 42 and 79 need rework. The n≥4 part might survive; D_n for n≥4 may indeed have boxicity n−1, and the counting argument might be repairable, but the paper does not currently prove it. The n=2 cases and most threshold-dimension statements are less affected; even for 3K2 the threshold dimension could be 2.

I want to give credit where it is earned: the definition of D(m), Lemmas 47 and 49, the partial-join lower-bound technique, and the clean surjectivity lemmas for rings are all worth keeping. The paper is not incoherent, and the authors are honestly engaging with the literature. But a load-bearing false claim sits at the center, and it is not a corner case: F2×F2×F2 is the standard example for reduced rings with three minimal primes.

Would I send this to referees? No, not in this state. The counterexample is too quick. I would tell the authors to repair the n=3 case, restate Theorem 8 with the correct exceptional behavior, and re-check every lower bound that uses box(D_n)=n−1. If that repair works, the paper will be valuable. As it stands, the main results are not citable.

Referee Report

1 major / 5 minor

Summary. The paper determines boxicity and threshold dimension of zero-divisor graphs for two classes of finite commutative rings: reduced rings and finite quotients of principal ideal domains. The main tool is a new family of combinatorial graphs, the integral covering graphs D(m), whose vertices are exponent vectors and whose edges encode coordinate-wise covering. The paper proves exact formulas for box(D(m)) and dim_TH(D(m)) (Theorems 42 and 43), uses these to describe the reduced zero-divisor graph of a finite reduced ring as the disjointness graph D(1^n), and then derives closed-form values for box(Γ(R)) and dim_TH(Γ(R)) in terms of the number of minimal prime ideals and the notion of a lonely prime ideal. For quotients of PIDs, analogous formulas are given in terms of the prime exponents m_i and the sizes of the residue fields. The paper also answers two questions posed by Chandran and Sahoo. I read the manuscript in good faith and stress-tested the n=3 base case of the combinatorial lower-bound machinery; the suggested counterexample that D(1^3) is 3K2 is not correct, because the singleton vertices {1}, {2}, {3} are pairwise disjoint and hence form a triangle. The manuscript's central theorems appear defensible, but the proof has a genuine gap at n=3 and a supporting lemma is false as stated.

Significance. If the results are correct, they provide exact answers to open questions in the boxicity literature and introduce a useful unifying gadget, the integral covering graph, that cleanly separates the combinatorial core from the ring-theoretic reductions. The algebraic reduction from zero-divisor graphs to D(1^n) and D(m) is carefully executed, with explicit surjectivity lemmas (Lemma 56 and Proposition 71) that establish the required isomorphisms. There is no circularity and no fitted parameters. However, the lower-bound engine for the n=3 case is not fully proved: the paper asserts without proof that box(D(1^3))=2, and the lemma one might use to prove it, Lemma 45, is false for n=3. Because Theorems 8, 42, 43, 79 and 80 all include n=3 cases, the manuscript needs a local but load-bearing repair before it can be accepted.

major comments (1)

[Section 3.1, Lemma 45 and Lemma 41] Lemma 45 is false as stated. For n=3, the interval supergraph of D(1^3) obtained from the singletons I_1=[0,5], I_2=[2,7], I_3=[4,9] and the 2-subsets I_{12}=[8,9], I_{13}=[5.5,6.5], I_{23}=[0.5,1] touches all three pairs {1,2}, {1,3} and {2,3}. The proof's line 'if {b,c} is in F, then |F|=3\le n-1' is only valid for n\ge 4. This matters because Lemma 41's treatment of n=3 currently relies on the unsupported assertion that D(1^3) 'is known to have boxicity 2'; the graph is the net graph (a triangle with a pendant edge at each vertex), and its boxicity is indeed 2, but the paper gives neither a proof nor a reference. Since the n=3 case is load-bearing for Theorems 8, 42, 43, 79 and 80, the manuscript must be revised to either prove box(D(1^3))=2 directly or cite a source, and Lemma 45 should be restricted to n\ge 4 or given a separate n=3 argument.

minor comments (5)

[Section 3.1, proof of Lemma 41] The sentence 'By Lemmas 44 and 46, each H_i deletes at most n-1+2 non-edges' appears to cite the wrong lemmas; the bound n-1 on touched pairs comes from Lemma 45, not from Lemma 44 alone. Please correct the citation.
[Section 3.1, Lemma 41] The claim that D(1^3) has boxicity 2 should be proved or explicitly referenced. As written, the proof for n=3 is an unverified assertion, and the paper's own Lemma 45 does not apply to n=3.
[Section 3.1, Lemma 45] Lemma 45 should be restated for n\ge 4. As stated, it is false for n=3, as shown by the interval representation in the major comment above.
[Figure 2] The caption 'D(13)' should read 'D(1^3)' to match the notation used in the text.
[Abstract] There is a typo in the first sentence: 'Thezero divisor graph' should be 'The zero divisor graph'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: combinatorial results are independently derived and ring theorems follow by explicit isomorphisms, not by fitting or self-reference.

full rationale

The paper's derivation chain is self-contained on the combinatorial side. The integral covering graph D(m) is defined purely combinatorially, and Theorems 42 and 43 (boxicity and threshold dimension of D(m)) are proved directly from interval/threshold graph representations, forbidden-subgraph characterizations, and explicit induced-subgraph constructions. No parameter is fitted to a subset of the data and then relabeled as a prediction; the lower and upper bounds are proven from the definitions. The ring-theoretic results are then connected to D(m) by explicit isomorphisms: Corollary 57 proves ΓE(R) ≅ D(1^n) using the surjectivity Lemma 56, which is itself proven from the prime-ideal property Lemma 17 and the characterization of zero divisors in reduced rings. Similarly, Corollaries 73–75 prove ΓE'(Q) ≅ D(m) using the independently proven surjectivity Proposition 71 and the valuation properties Propositions 68–69. Theorems 79 and 80 then follow by combining these isomorphisms with the already-established combinatorial theorems, together with Lemma 76's induced-subgraph lower bound and Lemma 47/84-type interval representations. The paper does cite prior work, including Chandran and Sahoo [8,9], but not as a load-bearing substitute for the arguments; the relevant ring-to-combinatorial reductions are reproved in the present text, and the external citations used (Roberts on boxicity, Chvátal–Hammer on threshold graphs, Bourbaki for commutative algebra facts) are standard and independent. The assertion that D(1^3) has boxicity 2, even if one disputed it, would be a mathematical correctness issue rather than a circularity, since it is not derived from the theorem it is used to prove and is not reduced to a fitted input. Overall, the claimed results are not equivalent by construction to their assumptions, and no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters; the paper is parameter-free combinatorics and algebra. The only new object is the integral covering graph D(m), which is an explicit definition rather than an unverified entity. Standard algebraic facts (PID factorization, prime ideal characterization of zero divisors) and standard graph parameter characterizations are imported from the literature.

assumptions (5)

standard math Unique prime factorization in principal ideal domains (Theorem 63)
Used throughout Section 5 to define exponent vectors a_i and to characterize divisibility in quotient rings (Propositions 65-68).
standard math In a reduced ring, the zero divisors are exactly the union of the minimal prime ideals (Corollary 21, from Bourbaki)
Basis for the coordinate map µ and Lemma 55 connecting products to prime membership.
standard math A graph has boxicity at most d iff it is the edgewise intersection of d interval graphs (Lemma 2, Roberts 1969)
Primary working definition of boxicity used in nearly every proof.
standard math Threshold graphs are exactly the graphs with no induced C4, P4, or 2K2 (Theorem 24, Chvatal-Hammer)
Used to prove Theorem 27 and in forbidden-subgraph arguments for threshold dimension.
standard math Boxicity is additive under joins (Lemma 23, Cozzens-Roberts)
Used for lower bounds via n-fold joins of K2.

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Pith. "Pith review of Boxicity and Threshold Dimension of Zero Divisor Graphs." pith.science (2026). https://pith.science/paper/BSHPWUO6

@misc{pith2026260827381,
  author       = {Pith},
  title        = {Pith review of: Boxicity and Threshold Dimension of Zero Divisor Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BSHPWUO6}},
  note         = {Machine review of arXiv:2608.27381}
}

abstract

The zero divisor graph $\Gamma(R)$ 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 $\Gamma(R)$ 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).

Figures

Figures reproduced from arXiv: 2608.27381 by the authors.

**Figure 2.** The graph D(13). 3.1 Proof of Lemma 41 We first handle the case n = 2. As m ̸= 12, we must have m1 ≥ 2 or m2 ≥ 2. Hence, (1, 0) and (0, 1) form two non-adjacent vertices of D(m), implying that dimTH(D(m)) ≥ box(D(m)) ≥ 1. By Proposition 37, it remains to show that box(D(1n)) ≥ n − 1 for n ≥ 3. The graph D(13), which is depicted in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Illustration of the construction of Hi for mi = 2 in the proof of Lemma 84. • We assign all v ∈ V with µi(v) = 2 and µn(v) = 0 to the interval [2, 4]. • We assign all v ∈ V with µi(v) = 0 and µn(v) = 1 to pairwise disjoint intervals contained within the interval (3, 4). • We assign all v ∈ V with µi(v) = 1 and µn(v) = 1 to the interval [1, 3]. • We assign all v ∈ V with µi(v) = 2 and µn(v) = 1 to the interval [0, 4]… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Illustration of the construction of H1 in the proof of Lemma 49. Corollary 48. Let n ≥ 3 and let m ∈ {1, 2} n \{2n}. Then box(D(m)) = n−1. Moreover, dimTH(D(1n)) = n − 1. Proof. The lower bound follows by Proposition 37. The upper bound follows from Lemma 47. Lemma 49.…

**Figure 5.** Figure 5: Illustration of the construction of Hi for i ∈ [n − 1] \ {1} in the proof of Lemma 49. • We assign all v ∈ V with vi = vn = 0 to pairwise disjoint intervals contained within (0, 1) if v1 ≤ 1, and within (2, 3) if v1 ≥ 2. • We assign all v ∈ V with vi = 0 and vn = 1 to …

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Works this paper leans on

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Pith tools

Reviewed August 28, 2026 · model on record in the stance chip above.

[1] [1]

Anderson and John D

David F. Anderson and John D. LaGrange. Commutative boolean monoids, reduced rings, and the compressed zero-divisor graph.Journal of Pure and Applied Algebra, 216(7):1626–1636, 2012. URL:https://www.sciencedirect.com/science/article/pii/S0022404911002660,doi:10.1016/ j.jpaa.2011.12.002

work page 2012

[2] [2]

Bellantoni, I

S. Bellantoni, I. Ben-Arroyo Hartman, T. Przytycka, and S. Whitesides. Grid intersection graphs and boxicity.Discrete Mathematics, 114(1-3):41–49, 1993. Combinatorics and algorithms (Jerusalem, 1988). doi:10.1016/0012-365X(93)90354-V. 23

work page doi:10.1016/0012-365x(93)90354-v 1993

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