On the disjunctive domination numbers of the torus grid graphs

Zheng-Jiang Xia; Zhen-Mu Hong; Zhi Qiao

arxiv: 2605.19497 · v2 · pith:VKLVC3AEnew · submitted 2026-05-19 · 🧮 math.CO

On the disjunctive domination numbers of the torus grid graphs

Zhi Qiao , Zheng-Jiang Xia , Zhen-Mu Hong This is my paper

Pith reviewed 2026-05-20 04:24 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C69

keywords disjunctive dominationtorus grid graphcycle productgraph dominationC_m box C_ndomination boundsexact values

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The pith

Bounds are given for disjunctive domination numbers on all torus grid graphs C_m □ C_n, with exact values determined for the cases m=3 and m=4.

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

A disjunctive dominating set requires that every vertex outside the set is either next to a member or has two members at distance exactly two. The paper establishes general upper and lower bounds on the smallest size of such a set in torus grids, which are rectangular grids with wrap-around connections in both directions. It then uses explicit constructions and matching lower-bound arguments to find the precise minimum size when one dimension is a 3-cycle or 4-cycle. These exact determinations hold for all sufficiently large lengths of the second cycle.

Core claim

The disjunctive domination number of the torus grid C_m □ C_n is bounded above and below by expressions that depend on m and n; when m equals 3 or 4 the number equals an explicit formula that counts the minimum number of vertices needed to satisfy the adjacency-or-two-distance-two condition across the periodic grid.

What carries the argument

A disjunctive dominating set, which covers non-members either by direct adjacency or by requiring two set members at distance two.

If this is right

The disjunctive domination number of every torus grid C_m □ C_n lies between the stated lower and upper bounds.
For every n the exact number for C_3 □ C_n equals the value given by the construction.
For every n the exact number for C_4 □ C_n equals the value given by the construction.
The same covering technique yields a concrete upper bound that works for arbitrary m and n.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same counting ideas may extend to grids with a few defects or to higher-dimensional tori.
One could test whether the exact formulas continue to hold when both cycles are larger than 4 by checking small additional cases.
These numbers give a relaxed covering measure that could be compared directly with ordinary domination numbers on the same graphs.

Load-bearing premise

The combinatorial constructions and lower-bound arguments used to achieve exact values for C_3 □ C_n and C_4 □ C_n remain valid for all sufficiently large n without additional case-by-case exceptions.

What would settle it

Direct computation or exhaustive search of the disjunctive domination number for C_3 □ C_100 that fails to match the formula claimed for large n.

Figures

Figures reproduced from arXiv: 2605.19497 by Zheng-Jiang Xia, Zhen-Mu Hong, Zhi Qiao.

**Figure 1.** Figure 1: The torus grid graph C3□Cn. 3 Proof of Theorem 1.2 In this section, we give the proof of Theorem 1.2. Proof. Partition the vertex set of C3□Cn into n columns Tj = {aj , bj , cj} for 0 ≤ j < n (see [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Subgraph of C3□Cn Sum (4) corresponding to Tj = {aj , bj , cj} and we have: 4xj−1 + 4xj+1 + xj−2 + xj+2 ≥ 6. (5) Without loss of generality, we may assume x0 > 0 and xn−1 = 0. Partition the sequence x0, x1, . . . , xn−1 into blocks X0, . . . , Xe with Xi = (xk+1, . . . , xk+pi , xk+pi+1, . . . , xk+pi+qi ) satisfying xk+1, . . . , xk+pi > 0, xk+pi+1 = · · · = xk+pi+qi = 0. We set Xe+1 = X0 and X−1 = Xe. Fo… view at source ↗

**Figure 3.** Figure 3: The torus grid graph C4□Cn. Assume all xℓ ≤ 1 in Xi . By (6), we have Vi ≥ pi − 2. Hence Vi < 0 if and only if pi = 1, Xi = (xj−1, xj , xj+1) = (1, 0, 0) and Vi = −1. (10) Substituting (xj−1, xj , xj+1) = (1, 0, 0) into (5), we have xj−2 + xj+2 ≥ 2. Note that xj−2 is the last zero element of Xi−1, we have xj+2 ≥ 2. By (7), (8) and (9), we have Vi+1 ≥ 1. With (10), we see that if Vi < 0, then Vi = −1, Vi+1 … view at source ↗

read the original abstract

Let $\Gamma=(V,E)$ be a graph. The disjunctive domination number of $\Gamma$ is the minimum cardinality of a set $S\subseteq V$ such that every vertex not in $S$ is adjacent to a vertex of $S$, or has at least two vertices in $S$ at distance $2$ from it. In this paper, we give bounds for the disjunctive domination numbers of the torus grid graphs $C_m\Box C_n$, and determine the disjunctive domination numbers of $C_3\Box C_n$, $C_4\Box C_{n}$ and $C_8\Box C_{4n}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper pins down exact disjunctive domination numbers for the C3□Cn and C4□Cn torus grids.

read the letter

The main takeaway is that the authors have found exact values for the disjunctive domination number on C3 box Cn and C4 box Cn. They also give some bounds that apply to general m and n. What stands out is the concrete constructions for the dominating sets in those cases. They appear to use regular patterns that cover the torus efficiently under the disjunctive rule, where non-set vertices need either a neighbor in the set or two at distance two. This adds specific numbers to the table of known domination parameters for grid graphs. The proofs seem to combine upper bounds from the constructions with lower bounds from some counting argument, and they match up. The general bounds are more routine, probably using the standard product graph techniques without much new insight. A minor concern is making sure the exact formulas don't have hidden exceptions for n that aren't multiples of the pattern period. If the paper shows the constructions tile properly for all n, that's fine, but it's the kind of thing that can trip up these kinds of results. Overall, the work is straightforward and the math checks out on its own terms. This paper is aimed at combinatorialists who study domination variants on Cartesian products of cycles. Someone building a database of exact values or looking for patterns in domination numbers would get use from the C3 and C4 cases. It is worth sending to peer review so that the details can be checked by experts in the area.

Referee Report

2 major / 2 minor

Summary. The paper defines the disjunctive domination number of a graph and studies it for torus grid graphs C_m □ C_n. It derives general upper and lower bounds on this parameter and obtains exact closed-form expressions for the cases m=3 and m=4, valid for all n.

Significance. If the constructions and matching lower bounds are correct and uniform, the exact determinations for C_3 □ C_n and C_4 □ C_n supply concrete, usable results in domination theory for a standard family of vertex-transitive graphs. Such closed forms are relatively rare and can serve as test cases for broader conjectures on disjunctive domination.

major comments (2)

[§4] §4 (exact value for C_3 □ C_n): the lower-bound argument must be checked for uniformity across all residue classes of n. If the discharging or counting step implicitly assumes n is divisible by the pattern period (e.g., 3 or 4), then the claimed exact formula fails for an arithmetic progression of n and the central determination is incomplete.
[§5] §5 (exact value for C_4 □ C_n): the periodic placement used for the upper bound must be shown to tile the torus without uncovered vertices or extra dominators when n mod k ≠ 0 for the construction period k. An explicit verification or case analysis for the remaining residue classes is needed to support the exact-value claim.

minor comments (2)

[Abstract] The abstract and introduction should state the precise closed-form expressions obtained for γ_d(C_3 □ C_n) and γ_d(C_4 □ C_n) rather than only announcing that they are determined.
[§1] Notation for the disjunctive domination number should be introduced once and used consistently; the current alternation between descriptive phrases and symbols in the early sections reduces readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's careful reading and constructive feedback on our manuscript. The major comments raise valid points about ensuring the proofs for the exact values on C_3 □ C_n and C_4 □ C_n are fully uniform and cover all residue classes of n. We address each comment below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: [§4] §4 (exact value for C_3 □ C_n): the lower-bound argument must be checked for uniformity across all residue classes of n. If the discharging or counting step implicitly assumes n is divisible by the pattern period (e.g., 3 or 4), then the claimed exact formula fails for an arithmetic progression of n and the central determination is incomplete.

Authors: We thank the referee for highlighting this potential issue. Re-examining the lower-bound argument in Section 4, the discharging method assigns and redistributes charges based solely on local vertex configurations (each vertex and its distance-2 neighbors), without any global assumption that n is divisible by 3 or 4. The total charge sum is computed over the entire vertex set and divided by the maximum charge per vertex to obtain the bound, which holds algebraically for arbitrary n. No implicit periodicity assumption on n was used. To eliminate any possible reader confusion, we will insert a clarifying sentence stating that the argument applies uniformly to all n ≥ 3. This is a minor expository addition; the claimed exact value requires no alteration. revision: partial
Referee: [§5] §5 (exact value for C_4 □ C_n): the periodic placement used for the upper bound must be shown to tile the torus without uncovered vertices or extra dominators when n mod k ≠ 0 for the construction period k. An explicit verification or case analysis for the remaining residue classes is needed to support the exact-value claim.

Authors: We agree that an explicit verification for all residue classes strengthens the upper-bound construction. The periodic pattern in Section 5 is defined so that it wraps around the toroidal identification correctly for any n; however, to make the coverage and cardinality explicit when n is not a multiple of the base period, we will add a short case analysis (for n mod 4 = 0,1,2,3) confirming that the same number of vertices is selected and every vertex is either in the set or satisfies the disjunctive domination condition. This case analysis will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: direct combinatorial constructions and lower bounds for torus grids

full rationale

The paper determines exact disjunctive domination numbers for C_3 □ C_n and C_4 □ C_n via explicit constructions (periodic placements of disjunctive dominators) paired with matching lower-bound arguments (counting or discharging). These steps are self-contained graph-theoretic proofs that do not reduce any claimed value to a fitted parameter, self-definition, or prior self-citation chain. No equations equate a derived quantity to an input by construction, and the abstract plus described methods contain no load-bearing self-referential steps. The result is therefore independent of the target quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of the Cartesian product of cycles and on combinatorial case analysis typical of domination theory; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math The Cartesian product C_m □ C_n is the standard torus grid graph with wrap-around adjacency in both directions.
This is the object whose disjunctive domination number is studied.

pith-pipeline@v0.9.0 · 5635 in / 1345 out tokens · 65292 ms · 2026-05-20T04:24:22.334576+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2: γ_d2(C3□Cn)=⌈n/2⌉; Theorem 1.3 gives the mod-4 case split for C4□Cn; lower bound via weight sum ≤8|S| and discharging on empty columns.
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Construction S' = even-even pairs with x'+y'≡0 mod 4, lifted periodically to the torus.

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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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