REVIEW 3 major objections 5 minor 5 references
Lattice Grids and Prisms are Antimagic
T0 review · 3 major / 5 minor · reviewed 2026-08-28 · deepseek-v4-flash
Pith's one-line read The paper proves that all lattice grid graphs and all prism graphs are antimagic, by constructing explicit edge labelings with pairwise distinct vertex sums.
desk verdict Solid new constructions for two open antimagic families; the proofs have fillable gaps that should be tightened before acceptance. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The mechanism for grids is a U/R marking of the path factor: incident edges receive different letters, then edges marked U are labeled across the layers in usual order while edges marked R are labeled in reversed order, so the restricted vertex sums $f_1^+$ and $f_2^+$ move in controlled opposing directions. A merged sequence $C$ interleaves all unused odd labels with the leftover even labels, keeping the $f_2^+$ differences large enough to dominate the possible drops in $f_1^+$. For prisms, the analogous mechanism is consecutive block labeling of the cycle layers plus a reversal modification on the second cycle layer when $n$ is even. The decomposition $f^+ = f_1^+ + f_2^+$ lets the proof compare vertex sums by comparing two monotone pieces.
What would settle it
Take the case $m=2,n=3$ for the grid $P_1[3] \times P_2[4]$, write out the paper's Phase 1 and Phase 2 labels, and compute every vertex sum; any equality between two vertex sums would disprove Theorem 1.1. A more targeted check computes the gap $f_2^+(u_2,v_{i+1}) - f_2^+(u_2,v_i)$ and the drop $f_1^+(u_2,v_{i+1}) - f_1^+(u_2,v_i)$ for each $i$, testing whether the asserted net growth of at least 2 actually holds.
Extended reading notes
Core claim
The central claim is Theorem 1.1 and Theorem 1.2: all lattice grid graphs and all prism graphs are antimagic. The construction splits each vertex sum into a contribution from the path or cycle copies and a contribution from the path copies that run across layers. For grids, edges inside path copies receive even numbers, while the remaining edges receive a merged sequence of odd numbers with a few leftover even numbers, and the proof orders the vertex sums in blocks separated by parity. For prisms, each cycle layer is labeled in consecutive blocks and the cross-layer path edges are oriented with a U/R marking; when the number of layers is even, one cycle layer is relabeled by reversal to restore the ordering. In both cases the vertex sums are shown to follow a strict total order, so they are pairwise distinct.
Load-bearing premise
The proof depends on monotonicity bounds for the partial vertex sums — for example, that along a row the path-layer part rises by at least four while the other part can fall by at most two — and these bounds are asserted rather than derived; the strict ordering used to separate the sums would break if any bound were off by one.
Editorial extensions
If this is right
- Every lattice grid graph $P_1[m+1] \times P_2[n+1]$ has an explicit antimagic labeling for all positive integers $m,n$.
- Every prism graph $C[m] \times P[n+1]$ has an explicit antimagic labeling for all $m \ge 3$ and $n \ge 1$.
- The two families join the growing list of Cartesian products known to satisfy the antimagic conjecture, extending beyond dense graphs and toroidal grids.
- Because the labelings are constructive, an antimagic labeling for any lattice grid or prism can be written down directly without searching.
Reading between the lines
- The same layer-by-layer strategy may extend to Cartesian products of more than two paths (higher-dimensional grids), since the partial-sum decomposition can be applied recursively; the paper does not claim this.
- The parity-dependent reversal used for even $n$ in prisms suggests that similar modifications could handle Cartesian products involving even cycles in other contexts.
- Making the asserted monotonicity gaps in Claim 3.1 explicit could yield a uniform minimum separation between vertex sums, which might simplify proofs for related product families.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves two theorems: every lattice grid graph P1[m+1] × P2[n+1] (the Cartesian product of two paths) is antimagic for integers m,n ≥ 1, and every prism graph C[m] × P[n+1] (the Cartesian product of a cycle and a path) is antimagic for m ≥ 3, n ≥ 1. The proofs are constructive. For grids, Phase 1 labels all edges in the P1-copies with the even numbers 2,4,...,2mn+2m using an alternating U/R orientation, and Phase 2 labels the P2-copies with a deliberately interleaved sequence of the remaining odd numbers and leftover even numbers. For prisms, Phase 1 labels the cycle copies blockwise and Phase 2 labels the path edges, with a reversal-and-renaming modification when n is even. In both cases, antimagicness is reduced to claims that the constructed vertex sums are strictly ordered.
Significance. If correct, Theorems 1.1 and 1.2 resolve two open problems left in Wang's paper on toroidal grids and provide explicit labelings for two natural classes of Cartesian-product graphs. The constructions are self-contained and parameter-free, and the base lemmas for paths and cycles are clean. The main proof, however, relies on several ordering assertions that are not fully justified, and one constancy claim in Claim 3.1 is false as stated. The central ideas appear sound, but the manuscript needs a rigorous repair of the ordering proofs before the results can be certified.
major comments (3)
- [Section 3.1, Claim 3.1] The proof of the second-row chain f^+(u2,v2) < ... < f^+(u2,vn) rests on the asserted inequalities f^+_2(u2,v_{i+1}) - f^+_2(u2,v_i) ≥ 4 and f^+_1(u2,v_{i+1}) - f^+_1(u2,v_i) ≥ -2 for i=2,...,n-1; neither inequality is derived, and a one-unit error in either bound could create collisions between adjacent terms in the claimed lexicographic order. Similarly, the transition f^+(u2,vn) < f^+(u3,v2) uses a bound on f^+_1 that is only sketched, with the case m=2 treated in a single line. Please supply complete derivations of these bounds before the proof can be accepted.
- [Section 3.1, Claim 3.1, final paragraph] The statement 'For the vertices of degree 4, clearly, f^+_1(ui,v2)=...=f^+_1(ui,vn) for i=3,...,m+1' is false for i=m+1. For that index, f^+_1(u_{m+1},v) is the single Phase-1 label on the edge ((u_m,v),(u_{m+1},v)), and the U/R orientation makes this label increase or decrease with v rather than remain constant; for example, it is increasing when the edge is U-oriented and decreasing when it is R-oriented. The ordering of the last row may still be provable by a suitable monotonicity argument, but as written the proof contains an incorrect assertion that must be replaced.
- [Section 4, Lemma 4.1] The proof of Lemma 4.1 defers the central ordering verification to 'It is easy to see' (for the reordered second C[m] component) and to 'straightforward to prove' (for the two complete chains of inequalities for f^+_1 and f^+_2). These chains are exactly what yields the final strict ordering f^+(u1,v1) < ... < f^+(um,v_{n+1}), and the modification for even n, which reverses labels and renames vertices, makes the verification non-obvious. Please provide a complete proof of both chains, covering all m ≥ 3, n ≥ 2, and both parities of n.
minor comments (5)
- [Section 3.1, after Phase 2] The bound 'Notice that 2t ≤ n-m' is used to conclude that only the last P2-copy may contain even labels, but the derivation of this bound is omitted; it follows from t ≤ (n-m)/2, which itself should be stated explicitly.
- [Section 3.1, definition of the sequence C] The merge of the odd sequence A and the even sequence B into C is described, but the paper does not explain why the particular interleaving is chosen; a sentence noting that after the first s-t terms the remaining odds and evens alternate and increase would help the reader follow the later ordering arguments.
- [Section 3.2, formulas for f^+_2] In the display of f^+_2(u1,vi) and f^+_2(u2,vi), the cases i=3,...,n are written as 4i-6 and 4i-4; the reader must check that the edge ordering used here is the same as in Lemma 2.1, and a brief reminder would improve clarity.
- [Figures 2-6] The typesetting of the vertex labels in the arXiv figures is garbled (for example, '1 1 ( , ) u v' instead of (u1,v1)); the final version should have readable labels.
- [References] Reference [5] is to a COCOON 2005 conference paper; if a journal version exists, it would be helpful to cite it as well.
Circularity Check
No significant circularity: the paper's proofs are self-contained explicit label constructions with direct distinctness arguments.
full rationale
The paper derives Theorems 1.1 and 1.2 by giving explicit bijective edge labelings and then proving, in Claims 3.1, 3.2, Lemma 4.1, and Lemma 4.2, that the induced vertex sums are pairwise distinct. There are no fitted parameters, no empirical inputs, and no quantity that is defined in terms of the claimed conclusion. The only external citations are to standard background results and to Wang's prior work on toroidal grids, but the labelings and ordering arguments are presented in the paper itself; Lemma 2.1 and Lemma 2.2 are proved directly, and Lemma 4.1 explicitly adapts the construction rather than importing it as an unverified premise. The manuscript does contain expositional gaps, such as the assertion in Claim 3.1 that f2(u2,v_{i+1}) - f2(u2,v_i) >= 4 and f1 differences are at least -2 without derivation, and the statement in Lemma 4.1 that the final ordering is 'straightforward to prove.' These are rigor or exposition concerns, not circularity: the needed inequalities are local arithmetic facts about explicitly defined labelings and do not presuppose the antimagicness conclusion. No step reduces by construction to its own input, and no load-bearing claim rests on a self-citation chain. Therefore the appropriate circularity score is 0.
Assumptions & free parameters
assumptions (2)
- standard math Standard definitions of graph, Cartesian product, and antimagic labeling as given in Section 2.
- standard math The non-standard edge representations for paths and cycles in Lemmas 2.1 and 2.2 are isomorphic to the usual path and cycle graphs.
Cite this review
Pith. "Pith review of Lattice Grids and Prisms are Antimagic." pith.science (2026). https://pith.science/paper/6NAJ4IOK
@misc{pith2026math0603106,
author = {Pith},
title = {Pith review of: Lattice Grids and Prisms are Antimagic},
year = {2026},
howpublished = {\url{https://pith.science/paper/6NAJ4IOK}},
note = {Machine review of arXiv:math/0603106}
}
abstract
An \emph{antimagic labeling} of a finite undirected simple graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called \emph{antimagic} if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every connected graph, but $K_2$, is antimagic. In 2004, N. Alon et al showed that this conjecture is true for $n$-vertex graphs with minimum degree $\Omega(\log n)$. They also proved that complete partite graphs (other than $K_2$) and $n$-vertex graphs with maximum degree at least $n-2$ are antimagic. Recently, Wang showed that the toroidal grids (the Cartesian products of two or more cycles) are antimagic. Two open problems left in Wang's paper are about the antimagicness of lattice grid graphs and prism graphs, which are the Cartesian products of two paths, and of a cycle and a path, respectively. In this article, we prove that these two classes of graphs are antimagic, by constructing such antimagic labelings.
Figures
Figures from the paper (3 more)
Reference graph
Works this paper leans on
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[1]
N. Alon, G. Kaplan, A. Lev, Y. Roditty and R. Yuster, Dense graphs are antimagic, Journal of Graph Theory, 47(2004), pp. 297-309
work page 2004
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[2]
J.A. Gallian, A dynamic survey of graph labeling, ninth edition, The Electronic Journal of Combinatorics, 5 (2005), DS6, pp. 1-148
work page 2005
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[3]
N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, INC., Boston, 1990 (Revised version, 1994), pp. 108-109
work page 1990
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[4]
Dan Hefetz, Anti-magic graphs via the Combinatorial NullStellenSatz, Journal of Graph The- ory, 50(2005), pp. 263-272
work page 2005
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[5]
Tao-Ming Wang, Toroidal Grids Are Anti-magic, Proc. 11th Annual International Computing and Combinatorics Conference COCOON’2005, LNCS 3595, Spri nger, 2005, pp. 671-679. 10
work page 2005
Reviewed August 28, 2026 · model on record in the stance chip above.
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