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

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Pith's one-line read The compressed zero divisor graph of Z_N has boxicity a−1 exactly for three prime-exponent patterns, and a in every other nontrivial case.

desk verdict Genuinely new exact result with a reusable C4-conflict lower-bound technique, but Lemma 4.9 has a misstated join that needs a one-line fix. read the letter →

arxiv 2608.23539 v1 pith:GYWZ66QY submitted 2026-08-24 cs.DM math.CO

L. Sunil Chandran , Suraj Kumar Sahoo This is my paper · ORCID
classification cs.DMmath.CO MSC 05C6213A99
keywords zerodivisorgraphscompressedgraphboxicitythresholddimensionintervaldisjointnessringofintegersmoduloN
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

The paper determines the exact boxicity of the compressed zero divisor graph of the ring of integers modulo $N$, a graph whose vertices are equivalence classes of zero divisors sharing the same annihilator. It proves that for $N$ with $a$ distinct prime factors, the boxicity is $a-1$ precisely when the exponent pattern is one of three types: one cubed prime and the rest square-free; all exponents $1$ with $a \geq 3$; or exponents only $1$ and $2$, with at least one of each. In all remaining cases with $a \geq 2$ the boxicity is $a$, except the single case $N = pq$ where the graph is a clique and boxicity is $0$. This settles the two open questions posed in the preceding paper on zero divisor graphs, and yields the boxicity of the disjointness graph of the power set of an $a$-element set.

What carries the argument

The central object is $\Gamma_E(\mathbb{Z}_N)$, the graph on annihilator-equivalence classes of zero divisors of $\mathbb{Z}_N$; in this ring the classes correspond to proper divisors of $N$, with adjacency meaning the product is divisible by $N$. The proof is carried by three mechanisms: the classical characterization (Theorem 2.1) that boxicity equals the minimum number of interval supergraphs whose edge intersection is the original graph; two lower-bound gadgets, the induced graph $aK_2$ obtained by removing a perfect matching from a complete graph on $2a$ vertices, and the $C_4$-conflict graph whose chromatic number forces boxicity from below; and, for the hardest upper bound, an explicit family of interval supergraphs indexed by the prime factors and built from the $p$-adic valuation function $f(u,q)$. The square-free case instead pushes boxicity down by forbidding certain induced subgraphs of interval graphs inside $\Gamma_E(\mathbb{Z}_N)$.

What would settle it

Take $N = p^3 q r$ for three distinct primes. Enumerate every pair of proper divisors $u,v$ of $N$ and check the interval assignments in Lemma 4.10: if any pair with $N \nmid uv$ overlaps in every interval graph $I_i$ and $I_j$, then the claimed upper bound $box = a-1$ fails. The same computation should verify Claims 1 and 2 for all pairs, not merely pictorially.

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Extended reading notes

Core claim

For $N = \prod_{i=1}^a p_i^{n_i}$ and $P = \{i : n_i = 1\}$, Theorem 1.7 states that $box(\Gamma_E(\mathbb{Z}_N)) = a-1$ if and only if (i) $|P| = a-1$ and the unique index outside $P$ has $n = 3$; (ii) $a \geq 3$ and $P = [a]$; or (iii) $\emptyset \subsetneq P \subsetneq [a]$ and every index outside $P$ has $n = 2$. The graph is a clique, with boxicity $0$, exactly when $a = 2$ and $n_1 = n_2 = 1$; all other cases with $a \geq 2$ have boxicity $a$. The same proof gives Theorem 1.8: the threshold dimension is $a-1$ only in the square-free case with $a \geq 3$, is $0$ for the complete graphs, and is $a$ otherwise. Consequentially, for any finite commutative reduced ring whose zero divisor graph has chromatic number $k \geq 3$, the compressed graph has boxicity $k-1$, so the zero divisor graph itself has boxicity between $k-1$ and $k$, and the lower bound is tight.

Load-bearing premise

The most delicate upper bound, in the case where $N$ has exactly one cubed prime factor and all others square-free, depends on the claim that the two interval graphs built by explicit formulas in Lemma 4.10 have exactly the adjacencies drawn and asserted in Figure 2; that claim is verified only by inspection.

Editorial extensions

If this is right

  • For any $N$, the boxicity of $\Gamma_E(\mathbb{Z}_N)$ can be read directly from the prime exponents of $N$, with no graph construction needed.
  • In the square-free case with $a \geq 3$, the disjointness graph of the non-empty proper subsets of $[a]$ has boxicity $a-1$, and even the subgraph on the 1-element and 2-element sets already forces boxicity $a-1$.
  • The open question on the tightness of the lower bound for boxicity of zero divisor graphs of reduced rings is settled: for chromatic number $k \geq 3$ the bound is $k-1$, not merely $\lfloor k/2 \rfloor$, and it is attained.
  • The threshold dimension of $\Gamma_E(\mathbb{Z}_N)$ is now exactly known in every case, matching the boxicity in the square-free and complete-graph cases and exceeding it by one elsewhere.
  • Adding the singleton sets to the disjointness graph of the 2-element subsets of $[a]$ raises its boxicity from $a-2$ to $a-1$.

Reading between the lines

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

  • The trichotomy suggests that boxicity is governed by the number of independent layers in the annihilator poset; a testable extension is whether an analogous exponent formula holds for other finite principal ideal rings, such as quotient rings of polynomial rings over finite fields.
  • The $C_4$-conflict graph coloring bound may be a general lower-bound technique for boxicity of algebraically defined graphs; one could check whether boxicity equals the chromatic number of the conflict graph for larger families than the modular rings treated here.
  • A concrete micro-test of the upper bound: for $N = p^3 q r$, locate which pair of non-adjacent vertices violates the interval representation if one perturbs the valuation-based interval lengths in Lemma 4.10, since the paper's proof of the two-cubed-primes case shows the $a-1$ representation must fail there.
  • Since the exponent pattern is a certificate of embedding dimension, algorithms that pre-process compressed zero divisor graphs could use the formula as an $O(a)$ structural test before attempting any geometric representation.
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Formalized claims in Lean

  1. Claim #1: For $N = \prod_{i=1}^a p_i^{n_i}$ and $P = \{i : n_i = 1\}$, Theorem 1.7 states that $box(\Gamma_E(\mathbb{Z}_N)) = a-1$ if and only if (i) $|P| = a-1$ and the unique index outside $P$ has $n = 3$; (ii) $a \geq 3$ and $P = [a]$; or (iii) $\emptyset \subsetneq P \subsetneq [a]$ and every index outside $P$ has $n = 2$. The graph is a clique, with boxicity $0$, exactly when $a = 2$ and $n_1 = n_2 = 1

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Editorial analysis

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

Referee Report

2 major / 3 minor

Summary. The paper determines the exact boxicity of the compressed zero divisor graph Γ_E(Z_N) for N = ∏_{i=1}^a p_i^{n_i}. Theorem 1.7 gives a complete dichotomy: box(Γ_E(Z_N)) = a−1 exactly in the three cases (one cubed prime with all other exponents 1, all exponents 1 with a ≥ 3, and cube-free numbers with at least one exponent 1 and all other exponents 2); box = 0 for a = 2 with both exponents 1; and box = a in all other cases with a ≥ 2. The paper also determines the threshold dimension (Theorem 1.8), answers an open question on reduced rings (Theorem 1.12), and derives corollaries on disjointness graphs of power sets. The proofs split according to the exponent pattern; lower bounds use Roberts graphs and a new C4-conflict-graph coloring argument, while upper bounds use explicit interval representations and isomorphisms to known cases.

Significance. If the result stands, it fully answers two open questions from Chandran and Sahoo's earlier paper and gives a clean structural dichotomoy for a natural algebraic graph class. The C4-conflict-graph method in Lemma 4.9 is a genuinely new lower-bound technique that goes beyond finding induced subgraphs of large boxicity, and the corollary that the disjointness graph of P([a]) has boxicity a−1 is elegant. The explicit interval assignments in Lemma 4.10 are a strength; I have verified Claims 1 and 2 and the three nonedge cases. The overall argument is coherent and the central claims are sound, but two printed subgraph identifications in the lower-bound proofs are incorrect as written and need local corrections.

major comments (2)
  1. [Section 4, Lemma 4.9, final paragraph] The claim that {N/q, N/q^2} ∪ (B\{N''}) induces K2 ∨ Γ_E(Z_N') is false. The two vertices N/q and N/q^2 are nonadjacent: their product is N'·q, which is not divisible by N = N'·q^2. The induced graph is instead the join of Γ_E(Z_N') with two universal nonadjacent vertices, i.e. (complement of K2) ∨ Γ_E(Z_N'). With the printed K2 reading, Observation 2.8 would give box = 0 + box(Γ_E(Z_N')) = |P|+2 = a−1, not the required lower bound a. With the corrected reading, the complement of K2 on two vertices has boxicity 1, so Observation 2.8 gives 1 + (|P|+2) = a, which restores the intended lower bound. This is a one-line correction, but as written the proof is internally inconsistent.
  2. [Section 4, Lemma 4.5] The verification states that S'_k ∪ S_i and S'_k ∪ S'_k' induce the graph 2K2. A direct exponent check shows that all four cross edges are present, so the induced graph is K_{2,2} (a C4), not 2K2. With the printed claim the union of the pairs would not form the Roberts graph aK2, and the lower bound box(Γ_E(Z_N)) ≥ a would not follow. Replacing '2K2' by 'C4' (or K_{2,2}) makes the intended Roberts-graph construction correct.
minor comments (3)
  1. [Section 4, Lemma 4.10] Claims 1 and 2 are asserted to be checkable by inspection. I did check them, and they are correct, but a brief indication of how the interval mapping enforces the adjacency conditions would improve readability and make the verification reproducible.
  2. [Section 4, Lemma 4.11] The formula N′ = 2·p_ℓ^{n_1}·N/(2^{n_1}·p_ℓ) is hard to parse. It would be clearer to describe the map as swapping the exponents of 2 and p_ℓ, and to state explicitly that this preserves Γ_E(Z_N) up to isomorphism.
  3. [Section 5, Proof of Theorem 1.8] In Claim 4 the sentence 'for r ∈ [a], S_r is an independent set' is followed by a classification of the induced graphs S_r ∪ S_{r'}. The logic that at most one S_r can be an independent set in a threshold supergraph is compressed; expanding it by one sentence would help.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: same-author black-box bounds are independent prior results, and the flagged Lemma 4.9 flaw is a correctness gap, not a circular reduction.

full rationale

The paper's derivation chain is not circular. The main external inputs, Theorem 1.4 and Theorem 3.1, come from the authors' earlier paper [21], but they concern box(Γ(Z_N)) and dim_TH(Γ(Z_N)) for the uncompressed zero divisor graph, with stated hypotheses that do not include the value of box(Γ_E(Z_N)). Their use is filtered through the valid induced-subgraph observation box(Γ_E(Z_N)) ≤ box(Γ(Z_N)), so no step reduces the target theorem to its own conclusion. The lower bounds are constructed directly from Roberts graphs, the C4-conflict chromatic bound, and divisor arithmetic, with no fitted parameter being renamed as a prediction; the case analysis of Section 4 verifies the trichotomy from explicit set constructions. The paper does answer two questions posed in [21], but answering one's own earlier open questions is not circular unless the answer is presupposed, and here the upper and lower bounds are established independently from the surrounding lemmas. The one flagged defect, in the last subcase of Lemma 4.9, is that the vertices {N/q, N/q²} ∪ (B\{N''}) are claimed to induce K2∨Γ_E(Z_N') even though N/q and N/q² are nonadjacent, so the printed lower-bound computation relies on the wrong supergraph; this is a verifiable correctness error in a construction, not a reduction of the theorem to its own statement. Because the proof leans heavily on same-author prior theorems in the upper-bound half, a low non-zero score is recorded, but no circular step is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard theorems in boxicity and threshold graph theory, plus the elementary divisor-class identification for Z_N. No free parameters are fitted and no new entities are postulated.

assumptions (6)
  • standard math Roberts' theorem: box(G) is the minimum number of interval supergraphs whose intersection is G (Theorem 2.1).
    Invoked throughout to formulate upper and lower bounds for boxicity.
  • standard math Threshold graphs are exactly the P4-free, C4-free, and 2K2-free graphs (Theorem 2.3).
    Used for threshold dimension results in Section 5 and Theorem 1.8.
  • standard math Erdős-Ko-Rado theorem for 2-subsets: the largest pairwise intersecting family of 2-subsets of [a] has at most a-1 sets.
    Used in Lemma 4.4 to force two disjoint pairs among many absent-edge pairs.
  • domain assumption In Z_N, Ann(x) = Ann(y) iff gcd(x,N) = gcd(y,N), so Γ_E(Z_N) is isomorphic to the induced subgraph of Γ(Z_N) on divisors D \ {1,N} (Observations 4.1 and 4.2).
    Core structural translation from ring theory to combinatorics; proved by elementary gcd arguments.
  • standard math Anderson-LaGrange lemma: for a reduced Noetherian ring R with k minimal prime ideals, Γ_E(R) is isomorphic to Γ(Z_2^k) (Lemma 1.10).
    Used to answer Question 1.6 for all finite reduced commutative rings.
  • standard math Cozzens-Roberts lower bound and Roberts graph boxicity (Theorem 2.11 and Observation 2.10).
    Used as the main lower-bound engine in several lemmas.

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Pith. "Pith review of The boxicity of the compressed zero divisor graph of the ring of integers modulo N." pith.science (2026). https://pith.science/paper/GYWZ66QY

@misc{pith2026260823539,
  author       = {Pith},
  title        = {Pith review of: The boxicity of the compressed zero divisor graph of the ring of integers modulo N},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GYWZ66QY}},
  note         = {Machine review of arXiv:2608.23539}
}
abstract

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 $\Gamma(R)$ for a ring $R $ is defined as the graph with the vertex set $V(\Gamma(R))=Z(R)$ and $E(\Gamma(R))=\{\{x,y\}\colon x,y\in Z(R)\text{ with }x\neq y\text{ and }x y=0\}$. One can define an equivalence relation $\sim$ on $V(\Gamma(R))$ 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 $\Gamma_E(R)$ for a ring $R$ is the simple graph obtained from $\Gamma(R)$ 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 $\Gamma_E(\mathbb{Z}_N)$. We show that when $a\geq 2$, $box(\Gamma_E(\mathbb{Z}_N))= a-1$ 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 $\Gamma_{E}(\mathbb{Z}_N)$ is a clique, and so, $box(\Gamma_{E}(\mathbb{Z}_N))=0$. In all other cases, $box(\Gamma_{E}(\mathbb{Z}_N))=a$.

Figures

Figures reproduced from arXiv: 2608.23539 by the authors.

Figure 2
Figure 2. A schematic diagram of the graphs Ii for i ∈ P\{l} (top), and the graph Ij (bottom). Each node is labeled with a set and represents a group of vertices from that set. Solid nodes represent a clique and hollow nodes represent independent sets. Edges are present between two nodes if and only if every vertex in one node is adjacent to every vertex in the other node. The edges between the vertices of a solid node and th… view at source ↗

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