{"id":"de108816-40a7-46b1-a51c-5fae0aac30e0","arxiv_id":"2412.06467","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove preservation of linear quotients under vertex duplication and expansion, exhibit a new infinite family of gapfree CDCC graphs whose edge ideal powers have linear quotients, and reduce a conjecture about all powers to checking finitely many powers.","lead":"This paper develops tools for proving that powers of edge ideals of gapfree graphs have linear quotients, a stronger property than linear resolution. It also constructs the first family of gapfree graphs containing cricket, diamond, C4 and C5 subgraphs whose edge ideal powers all have linear quotients from the second power onward.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.7 rests on the unverified finite base case in Proposition 5.4: the displayed 42-generator order of I(G)^2 is asserted with 'One can verify' and, if it fails, the whole induction and the CDCC family collapse.","rationale":"I read the paper in good faith. The structural framework—duplication (Prop. 2.3), expansion (Thm. 3.4), efficient orderings (Sec. 4), and the reduction to finitely many powers (Thm. 6.4)—is internally coherent, and I found no flaw in the long arguments of Prop. 5.4 beyond the base case. The main construction (Thm. 5.7) genuinely reduces to: (i) the base graph in Fig. 5 has I(G)^2 with linear quotients in the displayed order, and (ii) duplication preserves the property. (ii) is Proposition 2.3, whose proof I checked and found sound. (i) is asserted without demonstration. There are exactly 42 generators; checking linear quotients is a finite computation, but it is not a trivial one-liner because the order is long and non-obvious. Since the induction in Prop. 5.4 builds the orders for s≥3 from this base, a failure at a single colon condition destroys the family. The reader's weakest-assumption identifies the same type of gap, but I would narrow it: Proposition 5.2's analogous assertion is not needed for Theorem 5.7, so the truly load-bearing check is the Proposition 5.4 order. My verdict therefore stays CONDITIONAL: accept the structural claims, but require the finite certificate (or a computer-checkable verification) before Theorem 5.7 is fully supported.","tokens_in":25949,"tokens_out":21185,"duration_ms":184621,"concrete_test":"Write a short script (Macaulay2 or Python) that defines the graph of Figure 5 with edge labels e1=ab, e2=ap, e3=ax, e4=bx, e5=bq, e6=xp, e7=xq, e8=pz, e9=qz, constructs the 42 minimal monomial generators of I(G)^2 with the exact order displayed in Proposition 5.4, and checks the definition of linear quotients: for each i, the ideal generated by the first i-1 generators mod the i-th has colon ideal generated by variables. That is, for every pair i<j, there must be k<j with u_k:u_j a variable dividing u_i:u_j. Report the first violating pair if any. If the check passes, the base case is verified; if time permits, also build the efficient ordering N(3) from the verified N(2) and test I(G)^3 to confirm the induction starts.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 5.7: an infinite family of CDCC gapfree graphs whose edge ideal powers have linear quotients for all s≥2. Its proof is: Gamma_7 is obtained from the graph in Figure 5 by duplicating z, then Proposition 5.4 says I(Gamma_7)^s has linear quotients for all s≥2, and Proposition 2.3 propagates this under vertex duplication to all n≥7. Proposition 2.3 appears correct. The load-bearing step is therefore Proposition 5.4. That proposition proves the s≥3 case by induction from the base s=2, but the base is supplied only as a displayed order of 42 generators of I(G)^2 for the graph of Figure 5, with the sentence 'One can verify that I(G)^2 has linear quotients with respect to the following order'. No verification is shown, no certificate is given, and the induction proof depends on that order having the colon-ideal property at every position. If any single colon quotient (u_1,...,u_{i-1}):u_i is not generated by variables, the base fails and the induction step of Proposition 5.4 has no foundation. Proposition 5.2 contains the same pattern ('It is now a direct computation to verify...'), but it is not used in Theorem 5.7, so the decisive check is the order in Proposition 5.4. This is a missing finite verification in a load-bearing location, not a discovered error.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies powers of edge ideals of gapfree graphs and the stronger property of having linear quotients. After proving that linear quotients of powers are preserved under vertex duplication (Proposition 2.3) and under clique expansion when the resulting graph remains gapfree (Theorem 3.4), the authors introduce an explicit 'efficient ordering' (Conjecture 4.1) for passing from I(G)^q to I(G)^{q+1}. They verify this efficient ordering for the pentagon (Proposition 4.2), for two small graphs built from a pentagon plus a vertex (Propositions 5.2 and 5.4), and then use vertex duplication to obtain, for every n at least 7, a gapfree graph containing cricket, diamond, C4 and C5 as induced subgraphs whose edge ideal powers have linear quotients for all s at least 2 (Theorem 5.7). Section 6 develops a compatible-order framework and proves that if I(G)^2 through I(G)^q have linear quotients with compatible orders for some q at least 7, then every higher power has linear quotients (Theorem 6.4, with Propositions 6.5 and 6.6 covering the first two steps).","tokens_in":26229,"tokens_out":7650,"duration_ms":69252,"significance":"If fully validated, the paper makes two worthwhile contributions. First, Theorem 5.7 provides an unconditional infinite family of 'CDCC' gapfree graphs whose edge ideal powers have linear quotients from the second power onward, going beyond the chordal-complement and anticycle examples that were previously known. Second, Theorem 6.4 gives a genuine finite-reduction strategy for Conjectures 1.2 and 4.1: it shows that, under an admissible edge ordering and a compatible order for I(G)^2, verifying finitely many powers suffices for all higher powers. The paper contains no fitted parameters, uses standard external theorems, and its constructions and orderings are explicit. The main caveat is that the most important finite verification, the base order in Proposition 5.4, is currently asserted rather than demonstrated; this is a missing certificate in a load-bearing location rather than a discovered error.","major_comments":[{"comment":"The displayed 42-element order for I(G)^2 is introduced with the sentence 'One can verify that I(G)^2 has linear quotients with respect to the following order', but no verification is shown. This order is load-bearing for Theorem 5.7: Proposition 5.4's induction for s at least 3 starts from this base order, and Proposition 2.3 then propagates the property to all graphs Gamma_n with n at least 7. If any single colon ideal (w_1,...,w_{i-1}):w_i in this order is not generated by variables, the entire construction collapses. Please supply a complete verification: either a table of the relevant colon ideals or a machine-checkable certificate, such as Macaulay2 code, and state the base field over which the check is performed. The same issue appears in Proposition 5.2 ('It is now a direct computation to verify...'); although Proposition 5.2 is not used in Theorem 5.7, it should receive the same treatment.","section":"Proposition 5.4"},{"comment":"Theorem 6.4 is the centerpiece of the finite-reduction claim, but its proof contains several nontrivial subcases that are deferred rather than written out. Examples include the 'remainder of the proof follows similarly' in the z not equal to x subcase of Proposition 2.3, the 'One can verify that the same containment holds...' in Proposition 5.4, Case 2, and the 'similar argument' passages inside Claims 1 and 2 and Case 2.2 of Theorem 6.4. These are precisely the places where the compatible-order comparison is combined with the colon-ideal condition, so the current text does not allow the reader to audit the induction. Please expand these subcases fully or, where they are purely finite checks, provide a computer-verifiable certificate.","section":"Theorem 6.4"},{"comment":"Proposition 2.3 is used to pass from the graph Gamma_7 to the infinite family in Theorem 5.7, so its proof must be fully transparent. The subcase 'Suppose z not equal to x... The remainder of the proof follows similarly and it results with z in G(J)' is exactly the part of the proof where the ordering of the inserted duplicate monomials is used, and it is not immediate from the preceding displayed formulas. Please spell out this final subcase in detail; if this step is correct, doing so will also make the paper easier to verify.","section":"Proposition 2.3"}],"minor_comments":[{"comment":"The abstract and the introduction to Section 6 say that linear quotients of I^2, I^5, I^6 and I^7 imply linear quotients for all higher powers, but Theorem 6.4 requires I^2 through I^q for some q at least 7, and the proof of the full conclusion also uses Propositions 6.5 and 6.6. Please restate the hypothesis as I^2 through I^7 (or 'I^2, I^3, I^4, I^5, I^6 and I^7') to match the theorem.","section":"Abstract and Section 6"},{"comment":"In the displayed order for I(G)^2, the term listed as 'e3e8 = (ax)(qz)' should be 'e3e8 = (bx)(qz)', since e3 is the edge {b,x} and e8 is {q,z}.","section":"Proposition 5.2"},{"comment":"In the displayed order for I(G)^2, the term 'e3e9 = (ax)qz)' is missing a parenthesis; it should read 'e3e9 = (ax)(qz)'.","section":"Proposition 5.4"},{"comment":"The sentence 'Then there is u_l with l < t for which u_l : u_t is a variable and (u_l : u_t)|(u_l : u_t)' appears to contain a typo: the divisibility should be (u_l : u_t) | (u_i : u_t), not divisibility of the monomial by itself.","section":"Proposition 4.2, Case 1"},{"comment":"The graph on vertices {a,b,p,q,x,z} is said to be 'depicted below (Figure 2)', but the figure in the text is labeled 'Figure 3. Pentagon together with one vertex'. Please correct the cross-reference.","section":"Section 5, first paragraph"},{"comment":"The proof asserts without argument that duplicating a vertex of a CDCC graph again yields a CDCC graph. This is plausible, but a short justification that duplication preserves gapfreeness and retains the required induced subgraphs would make the proof self-contained.","section":"Theorem 5.7"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of math.AC and likely of interest to the commutative algebra community. The central risk is the unverified finite base order in Proposition 5.4; I would not reject on that basis, because a machine-checkable certificate or a table of colon ideals should be easy for the authors to supply. The Section 6 reduction is ambitious and needs the deferred case details filled in, but the overall direction is sound and the paper contains useful tools. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this paper. It does two genuinely useful things: it proves preservation of linear quotients under vertex duplication (Prop 2.3) and under clique expansion of a vertex in gapfree graphs (Thm 3.4), and it gives a finite-reduction theorem (Thm 6.4) that brings Nevo-Peeva closer to checkable form. It also constructs the first CDCC gapfree family—graphs with cricket, diamond, C4, C5—whose edge ideal powers have linear quotients from q=2 on (Thm 5.7). That is a real step in an active area.\n\nThe paper is honest: it discloses the concurrent preprint [5] for anticycles, and Remarks 6.7 and 6.8 point out where the argument stops. No fitted parameters, no circularity. The conjectures are stated as open, and the conditional implications are genuine.\n\nMy main concern is exactly where the stress test points: Proposition 5.4's base case. The s=2 order of I(G)^2 for the 9-edge graph of Figure 5 is asserted with 'One can verify', and Theorem 5.7 leans on it via duplication. If that order fails at any colon quotient, the family collapses. This is not a discovered error, but it is a missing finite certificate in a load-bearing place. Since the check is finite, the authors should either display the verification or provide a small Macaulay2/Sage certificate. Similarly, Proposition 5.2 says 'direct computation' but that one is not needed for the main theorem.\n\nThe other soft spot is Theorem 6.4: several subcases are passed with 'similar argument' and 'the proof goes similarly'. I found the main line convincing but those subcases deserve a close read, and the paper would be stronger if they were written out or at least clearly indexed. There are minor typos (e.g., e3e8 labeled (ax)(qz) in Prop 5.2's order), but they don't affect the arguments.\n\nWho is this for? Commutative algebraists working on monomial ideals and edge ideals. The paper deserves a serious referee rather than a desk reject. I would send it out and ask for the base-case verification and a tightening of Theorem 6.4 before final acceptance. If the finite checks are supplied, this becomes a good reference for the area.","headline":"A solid, honest paper with real new tools and one load-bearing finite check that should be supplied before the main family theorem is fully certified.","tokens_in":26820,"tokens_out":2360,"would_cite":true,"duration_ms":23128,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E40","13D02"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every n at least 7, there is a gapfree graph containing cricket, diamond, C4, and C5 as induced subgraphs whose edge ideal has linear quotients from the second power onward.","keywords":["edge ideal","gapfree graph","linear quotients","linear resolution","powers of edge ideals","CDCC graph","vertex duplication","admissible edge ordering"],"falsifier":"Take the displayed ordering of the 45 generators in Proposition 5.4 (or the 34 generators in Proposition 5.2), and for each generator w_i in that order compute the colon ideal (w_1,...,w_{i-1}):w_i; the claim holds only if every such colon ideal is generated by variables. A single colon ideal containing a non-variable generator, checkable by direct computation or a computer algebra system, would refute the theorem.","tokens_in":25749,"feed_emoji":"🕸️","tokens_out":9860,"duration_ms":97272,"temperature":0.7,"pith_summary":"This paper addresses an open conjecture about gapfree graphs: whether all sufficiently large powers of their edge ideals have linear resolutions. The authors work with the stronger property of linear quotients and conjecture that once a single power I(G)^q has linear quotients, every higher power does. They prove partial results, including that vertex duplication and gapfree clique expansion preserve linear quotients of powers, and that under compatible orderings only finitely many powers need to be checked. The main construction produces, for every n ≥ 7, a gapfree graph that contains cricket, diamond, C4, and C5 as induced subgraphs, yet all powers I(Γ_n)^s with s ≥ 2 have linear quotients and therefore linear resolutions.","feed_headline":"Even with C4, C5, cricket, diamond, edge-ideal powers stay linear","feed_subtitle":"New gapfree graphs on n≥7 vertices have linear quotients in every power from the second on.","key_machinery":"The central mechanism is the efficient ordering: given a linear-quotients order u_1 > ··· > u_r of the generators of I(G)^q, order the generators of I(G)^{q+1} as u_1e_1 > ··· > u_re_1 > u_1e_2 > ··· > u_re_s, deleting repeats, and iterate. The paper shows this ordering satisfies the defining colon condition for the pentagon and for the base graphs in Figures 3 and 5, with the base second-power verifications stated as direct computations. Section 6 introduces admissible edge orderings, total orders of edges in which any two disjoint edges force all edges incident to one endpoint of the larger edge to be larger than the smaller edge; these orderings allow the proof to reduce the persistence conjecture to checking finitely many powers, namely $I^{2}$ through $I^{7}$. Vertex duplication and clique expansion then propagate the base examples to infinitely many graphs.","core_discovery":"The paper constructs a family of CDCC graphs: gapfree graphs that have a cricket, a diamond, a 4-cycle, and a 5-cycle as induced subgraphs. For every n ≥ 7 there is such a graph Γ_n on n vertices for which every power I(Γ_n)^s with s ≥ 2 has linear quotients. Because a monomial ideal generated in one degree with linear quotients has a linear resolution, all these powers have linear resolutions as well. The construction starts from a seven-vertex base graph formed by adding a vertex to a pentagon with carefully chosen inner edges, verifies by explicit orderings that its second power has linear quotients, inductively extends those orderings to all higher powers, and then uses vertex duplication to reach every n ≥ 7. The paper also proves that duplication and gapfree expansion preserve linear quotients of powers, and that a compatible-order check through $I^{7}$ forces all higher powers to have linear quotients.","pith_inferences":["If the CDCC construction is correct, gapfree graphs with all powers from the second onward having linear quotients are not characterized by any finite list of forbidden induced subgraphs of size at most five, since this family deliberately contains every such small obstruction.","The efficient-ordering conjecture suggests that a single linear-quotients order for one power, rather than a separate argument for each exponent, is the right inductive structure; if the finite-check threshold could be lowered from seven to two, deciding Conjecture 1.2 for a given graph would reduce to checking I(G)^2.","Because the paper notes that duplicating both endpoints of an edge can make the matching number arbitrarily large while preserving gapfreeness, the CDCC family offers a testing ground for whether linear quotients of powers survive other graph operations that preserve gapfreeness."],"forward_implications":["For every n ≥ 7, there exists a gapfree graph on n vertices whose edge-ideal powers I^s have linear quotients for all s ≥ 2, so each such power has a linear resolution.","The previously studied sufficient condition that a gapfree graph avoid cricket, diamond, and C4 is not necessary: all three obstructions, together with C5, can be present while powers from the second onward remain well behaved.","Under the compatible-order framework, establishing linear quotients for just I^2 through I^7 would settle the persistence conjecture for all higher powers of a given graph.","Vertex duplication and gapfree clique expansion are general inheritance rules: any graph whose s-th power has linear quotients yields many new graphs with the same property, giving a flexible way to enlarge the CDCC family."],"supporting_citations":[{"why":"Poses the gapfree-graph conjecture about linear resolutions of large powers and supplies the negative example showing linear resolution can fail at q=2, which frames the paper's question.","marker":"[18]"},{"why":"Provides the definition of linear quotients and the standard implication that an equigenerated monomial ideal with linear quotients has a linear resolution, used throughout the paper.","marker":"[13]"},{"why":"Contributes the known regularity result that powers of edge ideals of gapfree graphs without cricket, diamond, or C4 have linear resolutions for all q ≥ 2, the baseline the CDCC family goes beyond.","marker":"[1]"},{"why":"Shows powers of edge ideals of (C4,2K2)-free graphs have linear resolutions, one of the results extended to graphs containing the forbidden induced subgraphs.","marker":"[9]"},{"why":"Extends the linear-resolution result for powers to the same obstruction-free class, providing the comparison point for the paper's new family.","marker":"[10]"},{"why":"Supplies the vertex-duplication technique for monomial ideals and graphs used in Proposition 2.3 to propagate linear quotients from one power to duplicated graphs.","marker":"[11]"},{"why":"The recent independent construction showing anticycles have linear quotients in all powers q ≥ 2, the class against which the paper positions its CDCC examples.","marker":"[5]"}],"fun_headline_variants":["Gapfree graphs with C4, C5, cricket, diamond still have linear quotients","Even with C4, C5, cricket, diamond, edge-ideal powers stay linear","Linear quotients survive C4, C5, cricket, diamond in gapfree graphs","All powers from second on have linear quotients for n≥7 gapfree graphs","Even with all four small subgraphs, edge-ideal powers keep linear quotients"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole induction rests on the unproved assertion in Propositions 5.2 and 5.4 that the two explicitly displayed orderings of the generators of I(G)^2 are linear-quotient orderings; the proofs say only that this follows by direct verification, so if either displayed ordering fails the colon condition at some position, the infinite family in Theorem 5.7 no longer follows.","fun_headline_variants_meta":{"raw":{"variants":["Gapfree graphs with C4, C5, cricket, diamond still have linear quotients","Even with C4, C5, cricket, diamond, edge-ideal powers stay linear","Linear quotients survive C4, C5, cricket, diamond in gapfree graphs","All powers from second on have linear quotients for n≥7 gapfree graphs","Even with all four small subgraphs, edge-ideal powers keep linear quotients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000706,"raw_usage":{"total_tokens":3182,"prompt_tokens":948,"completion_tokens":2234,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":2122}},"tokens_in":564,"tokens_out":2234,"duration_ms":17055,"temperature":1.0,"reasoning_tokens":2122,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T19:38:20.568448+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the displayed ordering of the 45 generators in Proposition 5.4 (or the 34 generators in Proposition 5.2), and for each generator w_i in that order compute the colon ideal (w_1,...,w_{i-1}):w_i; the claim holds only if every such colon ideal is generated by variables. A single colon ideal containing a non-variable generator, checkable by direct computation or a computer algebra system, would refute the theorem.","supporting_citations":[{"cited_title":"Nevo and I","cited_arxiv_id":null,"evidence_quote":"Poses the gapfree-graph conjecture about linear resolutions of large powers and supplies the negative example showing linear resolution can fail at q=2, which frames the paper's question."},{"cited_title":"Herzog and T","cited_arxiv_id":null,"evidence_quote":"Provides the definition of linear quotients and the standard implication that an equigenerated monomial ideal with linear quotients has a linear resolution, used throughout the paper."},{"cited_title":"Banerjee, The regularity of powers of edge ideals, J","cited_arxiv_id":null,"evidence_quote":"Contributes the known regularity result that powers of edge ideals of gapfree graphs without cricket, diamond, or C4 have linear resolutions for all q ≥ 2, the baseline the CDCC family goes beyond."},{"cited_title":"Erey, Powers of edge ideals with linear resolutions, Comm","cited_arxiv_id":null,"evidence_quote":"Shows powers of edge ideals of (C4,2K2)-free graphs have linear resolutions, one of the results extended to graphs containing the forbidden induced subgraphs."},{"cited_title":"Erey, Powers of ideals associated to ( C4,2K2)-free graphs, J","cited_arxiv_id":null,"evidence_quote":"Extends the linear-resolution result for powers to the same obstruction-free class, providing the comparison point for the paper's new family."},{"cited_title":"Francisco, H.T","cited_arxiv_id":null,"evidence_quote":"Supplies the vertex-duplication technique for monomial ideals and graphs used in Proposition 2.3 to propagate linear quotients from one power to duplicated graphs."},{"cited_title":"Powers of Edge Ideals with Linear Quotients","cited_arxiv_id":"2412.03468","evidence_quote":"The recent independent construction showing anticycles have linear quotients in all powers q ≥ 2, the class against which the paper positions its CDCC examples."}],"review_version":1}