{"id":"19bf608e-03e7-41cd-b283-33f9151cb208","arxiv_id":"2603.00752","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A composite ordering construction proves that certain modified anticycle graphs have edge ideals whose squares and cubes have linear quotients.","lead":"This mathematics paper develops a construction for ordering the generators of products of edge ideals so that they have \"linear quotients\", a strong algebraic regularity property. It uses this to show that a family of modified anticycle graphs has squared and cubed edge ideals with linear quotients.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.3's reduction to H_n uses a false automorphism: a↦n−2, b↦n sends the added edge {a+1,a} to {n−2,n−1}, whereas H_n adds {1,n}; so the proof does not cover every admissible (a,b) as written.","rationale":"The reader's weakest assumption identifies exactly this faulty automorphism, and my independent inspection of the text confirms it: the map a↦n−2, b↦n sends the added edge {a+1,a} to {n−2,n−1}, not to H_n's {1,n}, and the opposite orientation cannot be handled by the same rotation. Since Lemma 4.2 only proves the claim for H_n, Theorem 4.3 as written is not established for all a,b. This is a genuine gap in a load-bearing reduction, so the paper should not be accepted as-is. However, the gap appears fixable: for the a=n−2, b=n case, the reflection i↦(n−1)−i maps the modified graph to H_n, and rotations conjugate this to all a in the +2 orientation; the −2 orientation can likely be handled by a direct rotation to H_n. No evidence suggests the algebraic core (Lemmas 4.4, 4.12, 4.13) is wrong, though the unproved reduction (5.4) and the Lemma 5.1 statement/proof mismatch would also need repair. Thus the appropriate verdict is CONDITIONAL: the central claim is plausible and likely true, but the proof as written is incomplete.","tokens_in":20442,"tokens_out":19831,"duration_ms":187804,"concrete_test":"For n=7, take a=5, b=7 (so b=a+2). Compute E(H) after removing {5,7} and {6,1} from A_7 and adding {6,5}, and compare with E(H_7), which removes {5,7} and {1,6} and adds {1,7}. Enumerate all 14 dihedral automorphisms of A_7 and test whether any maps E(H) onto E(H_7); in particular test the rotation 5↦5,7↦7 (the identity, i.e. the map asserted in the proof) and the reflection i↦6−i. Then repeat for all admissible (a,b) for n=7 and n=8. If every H is isomorphic to H_n, Theorem 4.3 can be repaired by replacing the displayed automorphism; if some H is not isomorphic to H_n, the central claim is false.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central theorem (Theorem 4.3) claims that every graph H obtained from the anticycle A_n by removing {a,b}, {a+1,b+1} and adding {a+1,a} satisfies: I_H^2 and I_H^3 have linear quotients. Lemma 4.2 proves this only for the specific graph H_n defined in Section 3.2. The proof of Theorem 4.3 bridges the gap by asserting \"there is a graph isomorphism sending a↦n−2 and b↦n, so we may assume G=H_n.\" This is false for the stated map. In the orientation b≡a+2 (mod n), the rotation a↦n−2, b↦n sends the added edge {a+1,a} to {n−2,n−1}, while H_n is defined with added edge {1,n}; so the image of H is not H_n. In the opposite orientation b≡a−2, no automorphism of A_n can send a to n−2 and b to n while preserving adjacency, since automorphisms of A_n are exactly the dihedral symmetries of C_n and must preserve cyclic distance 2 or −2. Thus the displayed reduction is not justified. This is load-bearing because Lemma 4.2 is the entire proof of the infinite family; without a correct reduction, the theorem establishes only the single graph H_n. The gap appears repairable—for n=7, the reflection i↦6−i sends the a=5,b=7 case to H_7, and similar reflections work in general—but that argument is absent, and the b=a−2 orientation is not addressed. A separate unproved reduction, (5.4), is used repeatedly inside Lemma 4.13 and would also need independent checking, but the automorphism gap alone blocks the theorem's generality as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20916,"tokens_out":6312,"duration_ms":63618,"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":[{"comment":"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.","section":"Theorem 4.3 (proof)"},{"comment":"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.","section":"Lemma 5.1"},{"comment":"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.","section":"Lemma 4.13, Eq. (5.4)"}],"minor_comments":[{"comment":"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.","section":"Theorem 1.4"},{"comment":"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":"Theorem 4.3"},{"comment":"The definition says 'A graph is called edges if |V|<∞ and edges if u≠v...' — these should presumably be 'finite' and 'simple'.","section":"Section 2.1"},{"comment":"In the proof, 'we may assume that r≥6' should be 'n≥6'; the variable r is not introduced there.","section":"Remark 3.5"},{"comment":"The Macaulay2 output contains garbled entries such as 'x2x2x3x6' and 'x2x2_3x6'; these should be cleaned or replaced with a more readable transcript.","section":"Example 3.7"}],"recommendation":"major_revision","confidential_remarks":"The main construction and the overall strategy are promising, and the problems identified appear repairable: the automorphism reduction can likely be fixed with an explicit dihedral symmetry, Lemma 5.1 needs a corrected statement, and (5.4) needs a proof. However, because Theorem 4.3's generality and the completeness of Lemma 4.13's case analysis currently depend on these unproved points, the manuscript should not be accepted in its present form. Given the length and intricacy of the case analysis, a machine-checkable appendix or a fuller proof of (5.4) would substantially increase confidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: Lemma 4.4 is a genuinely useful tool for building linear quotient orderings of powers of sums of edge ideals, and Theorem 4.3's modified anticycle family is a real extension of the anticycle result in [2]. But the proof as written only establishes the case of the single graph H_n. The reduction in Theorem 4.3 claims any admissible (a,b) is isomorphic to H_n via a↦n−2, b↦n; that map sends the added edge {a+1,a} to {n−2,n−1}, whereas H_n adds {1,n}. The stress-test note is correct: the gap is load-bearing because Lemma 4.2 is the entire proof of the infinite family. The gap looks repairable — reflections of the cycle should handle the two orientations — but that argument is absent, and the b≡a−2 orientation is not addressed. As written, the theorem is only proved for H_n.\n\nWhat the paper does well: Lemma 4.4 is a clean sufficient condition — if each edge of G0 meets the star F0 and the mixed products I_{G0}^{s−j}I_{F0}^{j} have linear quotients, then the power of the sum does. The proof is short and convincing. The application to I^2_{H_n} via Lemma 4.10 and the lexicographic order on I_G I_F is explicit and checkable. The cube case is a long case analysis, but it is written out rather than hand-waved.\n\nSoft spots: besides the automorphism gap, the reduction (5.4) in Lemma 4.13 is asserted by analogy to (5.1) with no proof, and it is used repeatedly. Lemma 5.1's statement and proof also have a mismatch: the definition of 'agree in order t' does not quite align with the proof of (ii). Both are easily fixed, but they should be fixed. No circularity: the inputs are published results and the construction is from stated ingredients.\n\nWho this is for: people working on edge ideals, powers with linear quotients, and the Nevo–Peeva problem. The composite lemma deserves attention beyond this paper and may be reusable. The paper deserves a serious referee: the central idea is sound and likely correct, but the proof needs a corrected automorphism and a filled-in (5.4). I'd send it to peer review with instructions to focus on those two points; if they hold up, it's a publishable contribution to a niche but active area.","headline":"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.","tokens_in":21367,"tokens_out":2171,"would_cite":true,"duration_ms":22252,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13A70","13F55","05C25","05E40","05E45"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["linear quotients","edge ideals","anticycle","linear resolution","monomial ideals","star graphs","lexicographic order","composite linear quotient orderings"],"falsifier":"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}.","tokens_in":20323,"feed_emoji":"🔗","tokens_out":12412,"duration_ms":113735,"temperature":0.7,"pith_summary":"The paper proves that for a class of graphs obtained from an anticycle by deleting two edges whose endpoints differ by 2 modulo n and adding the edge joining the two smaller endpoints, the square and cube of the edge ideal have linear quotients, and hence linear resolutions. This gives infinitely many graphs satisfying part (iii) of a central open problem for powers two and three. The engine is a composite linear quotient ordering lemma: if a graph is a union of a subgraph and a star that meets every edge of the subgraph, an ordering of the s-th power of the edge ideal can be built by concatenating orderings of the mixed products of the two pieces. The paper then supplies such mixed orderings for an anticycle-star decomposition of the modified graph H_n, mostly via lexicographic order and a lengthy case analysis. The result is an explicit and total ordering of the minimal generators of I_H^2 and I_H^3.","feed_headline":"Squares and cubes of modified anticycles have linear quotients","feed_subtitle":"Removing two edges and adding one to an anticycle preserves linear-quotient behavior for square and cube powers.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Modified anticycles: squares and cubes keep linear quotients","New family of graphs with linear quotients in square and cube","Anticycle tweaks yield infinite graphs with linear-resolution powers","Square and cube of modified anticycles retain linear quotients"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Modified anticycles: squares and cubes keep linear quotients","New family of graphs with linear quotients in square and cube","Anticycle tweaks yield infinite graphs with linear-resolution powers","Square and cube of modified anticycles retain linear quotients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000724,"raw_usage":{"total_tokens":3022,"prompt_tokens":622,"completion_tokens":2400,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":366,"completion_tokens_details":{"reasoning_tokens":2342}},"tokens_in":366,"tokens_out":2400,"duration_ms":16693,"temperature":1.0,"reasoning_tokens":2342,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T19:52:27.083318+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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}.","supporting_citations":[],"review_version":1}