REVIEW 3 major objections 5 minor 11 references
Composite Linear Quotient Orderings of Ideals and Modified Anticycles
T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash
Pith's one-line read The paper proves that for infinitely many graphs obtained from anticycles by deleting two edges and adding one edge, the square and cube of the edge ideal have linear quotients, and hence linear resolutions.
desk verdict A genuinely useful composite linear quotient lemma and a plausible new infinite family of modified anticycles, but the main theorem's proof as written only establishes the special graph H_n because the stated automorphism reduction is false. 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 load-bearing construction is the composite linear quotient ordering lemma (Lemma 4.4): if H_0 = G_0 ∪ F_0, where F_0 is a star and every edge of G_0 is adjacent to some edge of F_0, and if I_{G_0}^{s−j} I_{F_0}^j has a linear quotient ordering for each j=0,...,s−1, then the concatenation of those orderings is a linear quotient ordering of I_{H_0}^s. In the application, G_0 is an anticycle A_{n−1} and F_0 is a star on n vertices whose leaves are a vertex cover of G_0. The individual orderings are obtained by lexicographic order, and the gluing step exploits the fact that the star's edge provides the variable that certifies the quotient condition. A second piece of machinery is the 'lexico
What would settle it
Compute the minimal generating set of I_H^3 for n=7 and (a,b)=(1,3), the graph that deletes {1,3} and {2,4} from A_7 and adds {1,2}, then test the linear-quotient condition of Lemma 3.1 on every pair of minimal generators; a single pair with no eligible predecessor monomial would refute the theorem, and the misstated automorphism can be checked directly by applying the stated map a→5, b→7 and observing that {1,2} goes to {5,6}, not to {1,7}.
Extended reading notes
Core claim
The paper's central claim is Theorem 4.3 (proved as Lemma 4.2): for n≥7, if a and b are vertices of the anticycle A_n with |a−b| congruent to ±2 modulo n, and H is formed by deleting the edges {a,b} and {a+1,b+1} from A_n and adding the edge {a+1,a}, then the square and cube of the edge ideal I_H have linear quotients. Because linear quotients force a linear resolution, this provides infinitely many graphs solving part (iii) of Problem 1.1 for s=2,3. The proof reduces to the normal form H_n, formed by deleting {n−2,n} and {1,n−1} from A_n and adding {1,n}, which decomposes as the union of an anticycle G=A_{n−1} and a star F with leaves 1,…,n−3, all centered at n. The argument then shows each
Load-bearing premise
The proof reduces every admissible graph in Theorem 4.3 to the normal form H_n by asserting a graph automorphism, but the automorphism as stated does not send the added edge to the edge {1,n} present in H_n; the reduction therefore needs a corrected automorphism for each admissible (a,b), and without it the proof does not cover all cases.
Editorial extensions
If this is right
- For every n≥7 and every admissible pair (a,b), the modified anticycle graph H satisfies that I_H^2 and I_H^3 have linear quotients, and hence linear resolutions.
- This yields an infinite family of graphs for which part (iii) of Problem 1.1 holds for s=2 and s=3, complementing the known fact that anticycles themselves have this property for all powers ≥2.
- The composite ordering lemma provides a general sufficient condition: any graph expressible as a star plus a subgraph whose edges all touch the star inherits linear quotient orderings for powers from the mixed ideals, provided those mixed ideals have linear quotients.
- Because the constructed orderings are explicit concatenations of lexicographic orders, they give a concrete way to exhibit linear resolutions for these edge ideals, not merely an existence statement.
Reading between the lines
- The proof of Theorem 4.3 contains a misstated reduction: the automorphism sending a to n−2 and b to n sends the added edge {a+1,a} to {n−2,n−1}, not to {1,n} as in H_n; a correct rotation or reflection exists for each admissible pair, but the paper does not supply it, so the reduction needs repair.
- The composite ordering lemma is a natural candidate for the kind of graph operation sought in Problem 1.2: attach a star that covers the graph; if the mixed ideals have linear quotients, the union has them for the corresponding power.
- The success of piecewise-lexicographic orderings here (lex fails globally on I_{H_n}^2, but works when split into a few intervals) suggests that testing piecewise-lex orders with few switches may be a useful general search strategy for linear quotient orderings of powers of edge ideals.
- One could test the s=4 case for the same family by trying the analogous decomposition into five mixed ideals; the case analysis would likely grow but the composite lemma still applies if the mixed orderings exist.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops a method for constructing linear quotient orderings of powers of edge ideals of graphs that decompose as G0 ∪ F0, where F0 is a star coning over a vertex cover of G0 (Lemma 4.4). It applies this machinery to a family of modified anticycle graphs H: for n ≥ 7, H is obtained from the anticycle A_n by removing two edges at cyclic distance 2 and adding the edge {a+1,a}. The central theorem (Theorem 4.3) claims that I_H^2 and I_H^3 have linear quotients, and hence linear resolutions, for every admissible pair (a,b). The proof reduces to a specific graph H_n and then gives explicit lexicographic-order case analyses for the mixed products I_G I_F, I_G I_F^2, and I_G^2 I_F, together with the composite ordering lemma. The paper also contains computational examples and remarks on obstructions to lexicographic linear quotients.
Significance. If the claims hold, this is a meaningful step toward Problem 1.1(iii): it produces an infinite family of non-cochordal graphs whose edge ideals have linear quotients in degrees 2 and 3, complementing the anticycle results of [2]. The composite ordering construction in Lemma 4.4 is a useful general tool, and the proofs are constructive, giving explicit orderings. The paper is honest about the computational cost of checking larger cases and gives some small Macaulay2 examples. However, the full generality of Theorem 4.3 currently rests on a reduction to H_n that is not correctly proved, and the long case analysis in Lemma 4.13 cites an unproved reduction and an incorrect statement of Lemma 5.1. These are repairable, but they block verification of the main theorem as written.
major comments (3)
- [Theorem 4.3 (proof)] The reduction 'there is a graph isomorphism sending a↦n−2 and b↦n' is not correct as stated. If b≡a+2 (mod n), the rotation sending a to n−2 and b to n sends the added edge {a+1,a} to {n−2,n−1}, whereas H_n has added edge {1,n}; the image is not H_n. If b≡a−2, no automorphism of A_n can send a to n−2 and b to n while preserving the cyclic distance, since automorphisms of the anticycle are dihedral symmetries of C_n. Thus Lemma 4.2, as written, proves the theorem only for the single graph H_n. A correct dihedral automorphism argument covering both orientations is needed. This is load-bearing: it is the only bridge from Lemma 4.2 to the infinite family claimed in Theorem 4.3.
- [Lemma 5.1] Statement (ii) asserts M1 >lex M3, but the proof establishes M3 >lex M2, and M1 >lex M3 is false in general. Example: in S=K[x1,x2,x3], take s=2, M1=x3x1 and M2=x2x1. Then t=0, j=1, and M3=(x3/x1)M2=x3x2, while x3x2 >lex x3x1, contradicting M1 >lex M3. Part (iii) is then based on this false claim. Since Lemma 5.1(iii) is cited repeatedly in Lemmas 4.12 and 4.13, the statement must be corrected: the needed property is that M3 precedes M2 (i.e., M3 >lex M2), together with the divisibility condition. Each application must be rechecked against the corrected statement.
- [Lemma 4.13, Eq. (5.4)] The reduction (5.4) — that the lemma holds if (i,l)=(a,b), (i,l)=(c,d), (j,r)=(a,b), or (j,r)=(c,d) — is asserted without proof, with only 'By a similar proof to that of (5.1)'. It is used many times in the subsequent case analysis (e.g., 'we are done by (5.4)' and 'By (5.4) we can assume...'). This is not a harmless remark: the entire case breakdown for Lemma 4.13 depends on excluding these configurations. A complete proof of (5.4), or a citation to a proved analogue, is required before the case analysis can be verified.
minor comments (5)
- [Theorem 1.4] The statement says 'Then I_G^2 and I_G^3 have linear quotients' but no graph G has been defined; it should be I_H^2 and I_H^3.
- [Theorem 4.3] Typo: 'positive reside mod n' should be 'positive residue mod n', and the symbol c is used both for the residue and in the congruence.
- [Section 2.1] The definition says 'A graph is called edges if |V|<∞ and edges if u≠v...' — these should presumably be 'finite' and 'simple'.
- [Remark 3.5] In the proof, 'we may assume that r≥6' should be 'n≥6'; the variable r is not introduced there.
- [Example 3.7] The Macaulay2 output contains garbled entries such as 'x2x2x3x6' and 'x2x2_3x6'; these should be cleaned or replaced with a more readable transcript.
Circularity Check
No significant circularity: the central construction is self-contained and rests on independent external results; the flagged proof gaps are correctness issues, not circular derivation.
full rationale
The paper's derivation chain is not circular. Lemma 4.4 is a conditional construction: it assembles a linear quotient order for I^s_{G0∪F0} from independently supplied linear quotient orders of I^{s-j}_{G0}I^j_{F0}. The application to H_n discharges each condition: I^2_G and I^3_G use Theorem 5.4 [2]/Theorem 2.2 [2], I_F and I^2_F are shown directly for the star, I_G I_F by Lemma 4.10, and the two mixed terms for the cube by Lemmas 4.12–4.13, all verified in the text (modulo the unproved sub-reduction (5.4)). No fitted parameter is relabeled as a prediction, and no definition of H_n or of the modified family encodes the target conclusion. The theorem's reduction to H_n via 'there is a graph isomorphism sending a→n−2 and b→n' is asserted rather than proved, and the skeptic's note shows the displayed map can fail; however, that is a correctness gap in the reduction, not a circular derivation. Similarly, the invocation of (5.4) as 'by a similar proof' is an omitted case analysis, not a self-referential use of the theorem. Independent external inputs (Herzog–Hibi–Zheng, Fröberg, [2]) are cited rather than re-derived from the target, so the central claim retains independent content. The result should therefore be scored as free of circularity, while its proof carries substantial correctness risk outside the scope of this circularity analysis.
Assumptions & free parameters
assumptions (5)
- standard math Herzog-Hibi-Zheng theorem: if a quadratic monomial ideal has linear quotients, then all its powers have linear quotients.
- standard math Fröberg's theorem: an edge ideal has linear resolution iff it has linear quotients iff its graph is cochordal.
- domain assumption Theorem 2.2 of [2]: I^s_{A_m} has linear quotients for m≥5, s≥2.
- ad hoc to paper There exists a graph automorphism of A_n sending any admissible modified anticycle H to H_n.
- domain assumption Ground field K is arbitrary and the linear quotients property is characteristic-independent.
Cite this review
Pith. "Pith review of Composite Linear Quotient Orderings of Ideals and Modified Anticycles." pith.science (2026). https://pith.science/paper/BI4MB5HF
@misc{pith2026260300752,
author = {Pith},
title = {Pith review of: Composite Linear Quotient Orderings of Ideals and Modified Anticycles},
year = {2026},
howpublished = {\url{https://pith.science/paper/BI4MB5HF}},
note = {Machine review of arXiv:2603.00752}
}
read the original abstract
In this paper we give a construction for a linear quotient ordering of a class of products of two ideals which have linear quotients. We apply this construction to give a class of modified anticycle graphs whose square and cube have linear quotients.
Reference graph
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