{"id":"28fde592-d927-4ee0-9865-4c71db397202","arxiv_id":"2506.01810","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every k≥2, a Cohen-Macaulay very well-covered whiskered bipartite graph has non-linear HS_k, and the paper identifies classes where HS_k does have linear quotients.","lead":"The paper builds graphs whose vertex cover ideals have homological shift ideals without linear resolutions, disproving a published conjecture and several theorems. It then proves that this failure does not occur for Cohen-Macaulay chordal graphs, Cameron-Walker graphs, and some clique corona graphs.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central counterexample is internally valid; the unproved very well-covered property of G_k is true and readily verified.","rationale":"I traced the central counterexample in detail. The graph G_k = W(C_{2k}) is a clique-whiskered graph, and its minimal vertex covers are in bijection with vertex covers S of the cycle C_{2k}, via C = {x_i: i in S} union {y_i: i not in S}. For a shift of size k, the set sigma must be disjoint from C and lie in the set of non-C vertices, which has size 2k-|S|. Since C_{2k} has vertex-cover number k, the only way to have |sigma|=k is to take S to be one of the two minimum vertex covers of C_{2k}, namely the odd or even vertices. This yields exactly the two monomials M*(y1 y3 ... y_{2k-1}) and M*(y2 y4 ... y_{2k}), where M is the product of all x_i. Their ideal is generated in degree 3k and has resolution given by the Koszul complex on two monomials with lcm equal to the product of all 4k variables, so its regularity is 4k-1. Since 4k-1 > 3k for k>=2, the ideal is not linear and hence does not satisfy linear quotients. The very well-covered property, flagged by the reader, is true: every maximal independent set contains exactly one vertex from each pair {x_i,y_i}, all have size 2k, and |V|=4k=2alpha. The proof should include this verification, but its absence is a presentation gap, not a correctness flaw. The long proof of Theorem 2.11 and the sketch of Theorem 2.17 are not load-bearing for the central counterexample. Therefore I find no significant objection to the main claim.","tokens_in":15474,"tokens_out":33531,"duration_ms":326680,"concrete_test":"Verify the two load-bearing facts directly: (1) In G_k=W(C_{2k}), confirm that every maximal independent set has size 2k, so alpha=2k and |V|=4k=2alpha, making G_k very well-covered; (2) For k=4, compute the multigraded Betti table of J(G_4) with Macaulay2 and check that HS_4(J(G_4)) is exactly (prod_{i=1}^8 x_i)<y1 y3 y5 y7, y2 y4 y6 y8>, generated in degree 12 with regularity 15; if so, Theorem 2.7's counterexample is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The only candidate load-bearing concern is the reader's observation that the very well-covered property of G_k is asserted but not proved in the body. This concern does not land: G_k is the whiskered graph on C_{2k}, and every maximal independent set contains exactly one vertex from each pair {x_i, y_i}, so all maximal independent sets have size 2k; hence alpha(G_k)=2k, |V(G_k)|=4k=2alpha, and G_k has no isolated vertices, so G_k is very well-covered. The core computation of Theorem 2.7 also checks out: minimal vertex covers C correspond to vertex covers S of C_{2k} via C={x_i: i in S} union {y_i: i not in S}; the condition |sigma|=k with sigma disjoint from C forces S to be one of the two minimum vertex covers of C_{2k}, giving exactly the two generators (prod x_i)(prod odd y_i) and (prod x_i)(prod even y_i). These two monomials have lcm equal to the product of all 4k variables, so their ideal has regularity 4k-1 while being generated in degree 3k; for k>=2 this is non-linear, so HS_k(J(G_k)) cannot have linear quotients. The contradiction with the quoted theorems is therefore valid provided the quotations are accurate. No load-bearing mathematical objection remains.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the kth homological shift ideals HS_k(J(G)) of the vertex cover ideal of a graph G. The main result is a counterexample: for each k ≥ 2, the whiskered graph G_k over the even cycle C_{2k} is a Cohen-Macaulay bipartite (and very well-covered) graph for which HS_k(J(G_k)) fails to have a linear resolution, and hence fails to have linear quotients. The explicit computation identifies HS_k(J(G_k)) with (∏_{i=1}^{2k} x_i)·(y_1y_3⋯y_{2k−1}, y_2y_4⋯y_{2k}), whose regularity is 4k−1 while it is generated in degree 3k. This is claimed to contradict several published results and a conjecture. The paper also proves positive results: for Cohen-Macaulay chordal graphs, HS_k(J(G)) has linear quotients for all k; for Cohen-Macaulay Cameron-Walker graphs and for clique corona graphs with all attached cliques of size at least 2, HS_k(J(G)) is weakly polymatroidal (hence has linear quotients).","tokens_in":15644,"tokens_out":21000,"duration_ms":207142,"significance":"If correct, the counterexample is significant: it disproves published theorems of Crupi and Ficarra and a conjecture, showing that the linear quotient property is not preserved by homological shifts even for Cohen-Macaulay very well-covered graphs. The construction is transparent and the key computation is explicit, so the negative result is convincing. The positive theorems identify nontrivial classes where the property is preserved; the proof for chordal graphs is a substantial inductive argument using Betti splittings. However, the proofs of the Cameron-Walker and clique-corona theorems contain gaps that need to be repaired before the full set of claims can be accepted.","major_comments":[{"comment":"In the proof of Theorem 2.14, the case where G is a disjoint union of K2 or K3 components is dismissed with the sentence 'then G is a chordal graph, and thus, the result holds by Proposition 2.4.' Proposition 2.4, however, only establishes that J(Gπ) has linear quotients; it does not imply that HS_k(J(G)) is weakly polymatroidal, which is the property asserted in Theorem 2.14. The chordal case therefore needs a separate verification (or the statement of Theorem 2.14 must be weakened to linear quotients, which would already follow from Theorem 2.11).","section":"Section 2, proof of Theorem 2.14"},{"comment":"The proof of Theorem 2.17 is a single sentence: 'repeating the same argument up to the Subcase-I of the proof of Proposition 2.14.' This is not sufficient as a proof, since the setup differs from that of Theorem 2.14: the base graph Γ is arbitrary, and each attached clique has size t_i+1 >= 3 rather than being a triangle attached to a bipartite graph. In particular, the vertex-exchange construction in Subcase-I (replacing w_{i2} by v_i in a minimal vertex cover) relies on the triangle having exactly three vertices and must be re-checked for larger cliques. Please expand the proof or explicitly state the modified exchange argument.","section":"Section 2, Theorem 2.17"}],"minor_comments":[{"comment":"The very well-covered property of G_k is asserted in the abstract and introduction but never proved in the body. It is needed to invoke [7, Theorem 4.1]; please add a one-sentence proof, e.g., every maximal independent set contains exactly one vertex from each pair {x_i,y_i}, so all maximal independent sets have size 2k = |V(G_k)|/2.","section":"Abstract and Section 2 (Theorem 2.7)"},{"comment":"There are several numbering/reference inconsistencies: the proof of Theorem 2.7 cites 'Proposition 2.6' but the result is Remark 2.6; Remark 2.6 refers to 'Proposition 2.5' but the statement is Corollary 2.5; the proof of Theorem 2.14 cites 'Proposition 2.13' for the classification of Cohen-Macaulay Cameron-Walker graphs, which is Theorem 2.13; and the introduction refers to 'Theorem 2.7' as a 'Proposition' in one place. Please unify the cross-references.","section":"Throughout Section 2"},{"comment":"The regularity computation for HS_k(J(G_k)) = (∏ x_i)·(y_odd, y_even) is attributed to [30, Lemma 8]; it would be helpful to include the short direct computation (the ideal (y_odd, y_even) has resolution 0 → R(−2k) → R(−k)^2, so multiplication by ∏ x_i shifts the regularity to 4k−1), making the proof self-contained.","section":"Section 2, proof of Theorem 2.7"},{"comment":"The statement 'and thus, linear quotients for all k≥0' should read 'and thus, has linear quotients for all k≥0'.","section":"Section 2, Theorem 2.17 statement"}],"recommendation":"major_revision","confidential_remarks":"The counterexample is sound, and the authors' computation checks out. The main risk is the accuracy of the quotations from [7] and [8]; since the paper's central claim is a contradiction with those papers, the authors should re-verify the exact hypotheses of [7, Theorem 4.1] and [7, Conjecture 4.4] and state them explicitly. The gaps in Theorems 2.14 and 2.17 are repairable, but they are real as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the counterexample is correct and significant, and the positive results for chordal and Cameron–Walker graphs look sound; the clique corona section is underproved as written. This deserves a serious referee.\n\nThe new thing is the family G_k = W(C_{2k}). For each k ≥ 2 it is Cohen–Macaulay, bipartite, whiskered, and very well-covered, and the paper proves HS_k(J(G_k)) = (∏ x_i)(y_odd, y_even). That ideal is generated in degree 3k and has regularity 4k−1, so it cannot have a linear resolution or linear quotients. That contradicts several published results and a conjecture, assuming the quotations in Section 1 are faithful. I checked the computation; it is clean. The very-well-covered property is asserted but never proved; the stress test is right that it is true (all maximal independent sets have size 2k). The authors should add the one-line proof, but this is not a flaw in the mathematics.\n\nThe chordal theorem is the real technical content. The ordering in (2.2) plus Lemma 2.10 is a genuinely useful device, and the colon-ideal arguments are dense but convincing. The Cameron–Walker proof via weak polymatroidality also looks correct. The clique corona theorem (t_i ≥ 2) is a different matter: the proof just says \"repeating the same argument\" and refers to the wrong numbered statements. A referee would have to reconstruct the whole argument. That section needs to be written out.\n\nThere are also many cross-reference slips: Theorem 2.7 gets called Proposition 2.7, Remark 2.6 becomes Proposition 2.6, Theorem 2.14 becomes Proposition 2.14, and so on. Confusing but harmless.\n\nThe one load-bearing external dependency is the accuracy of the citations of [7, Theorem 4.1, 4.2, Corollary 4.11], [8, Theorem 4.8], and [7, Conjecture 4.4]. If those statements are what the authors say they are, the contradiction is real. A referee must verify them.\n\nRecommendation: accept for peer review. The counterexample alone is enough to justify referee time; the chordal and Cameron–Walker results are likely publishable once the clique corona proof is filled in and the internal references are fixed.","headline":"The counterexample is real and important; the chordal/Cameron–Walker proofs are solid, but the clique corona section and internal references need work.","tokens_in":16200,"tokens_out":7261,"would_cite":true,"duration_ms":68157,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D02","05E40","13F55","13H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For each k≥2, a Cohen-Macaulay bipartite whiskered graph whose kth homological shift ideal lacks a linear resolution, contradicting published theorems and a conjecture; positive results cover chordal, Cameron-Walker, and certain clique…","keywords":["homological shift ideals","linear quotients","vertex cover ideals","Cohen-Macaulay graphs","chordal graphs","Cameron-Walker graphs","clique-whiskered graphs","weakly polymatroidal ideals"],"falsifier":"Compute, for k=2, the minimal free resolution of the ideal (x_1x_2x_3x_4)⟨y_1y_3, y_2y_4⟩ in K[x_1,...,x_4,y_1,...,y_4] using any computer algebra system; if the Castelnuovo-Mumford regularity is 6 (the generation degree) rather than 7, then the ideal would have a linear resolution and the counterexample would fail. Alternatively, verify the very well-covered property of the whiskered 4-cycle by checking |V|=2α(G) and that every maximal independent set has size α.","tokens_in":15213,"feed_emoji":"","tokens_out":8619,"duration_ms":73648,"temperature":0.7,"pith_summary":"The paper studies when the kth homological shift ideal of a vertex cover ideal inherits a linear resolution from the cover ideal itself. Its central result is a counterexample: for each k≥2, there is a Cohen-Macaulay bipartite whiskered graph G_k, also very well-covered, for which HS_k(J(G_k)) fails to have a linear resolution, and therefore also fails the stronger linear quotient property. This contradicts four published theorems and a conjecture about such shift ideals. The paper then identifies positive classes where the property does hold: Cohen-Macaulay chordal graphs have linear quotients for all k, and Cohen-Macaulay Cameron-Walker graphs as well as certain clique corona graphs are weakly polymatroidal, which implies linear quotients. The upshot is a precise map of which Cohen-Macaulay graph classes preserve the linear-resolution behavior under homological shifts.","feed_headline":"Whiskered even cycles defeat shift-ideal property","feed_subtitle":"For each k≥2, the kth homological shift ideal of such a graph lacks linear resolution, contradicting earlier results.","key_machinery":"The key machinery is a combinatorial description of the minimal generators of HS_k(J(G_π)) for a clique-whiskered graph G_π, developed in Propositions 2.4 and 2.6: every generator is x^C x^σ, where C is a minimal vertex cover and σ is a k-element subset of the union of the neighbor sets of the whisker vertices not contained in C, with an explicit formula for the colon sets that certify linear quotients of J(G_π). Applied to G_k = W(C_{2k}), where the clique partition π consists of the singleton cycle vertices, this description forces σ to be exactly one of the two alternating sets {x_1,x_3,...,x_{2k-1}} or {x_2,x_4,...,x_{2k}}, yielding the closed-form ideal above. The non-linear resolution then follows by comparing the generation degree 3k with Woodroofe's regularity calculation reg = 4k-1. For the positive results, the machinery is an inductive Betti-splitting decomposition of HS_k(J(G_π)) together with a specially designed ordering of minimal generators that satisfies a replacement property (∗), plus the Kokubo–Hibi notion of weakly polymatroidal ideals.","core_discovery":"The central claim is that for every integer k≥2, the graph G_k obtained by attaching a leaf to each vertex of a 2k-cycle—a whiskered graph, hence Cohen-Macaulay and very well-covered—has the property that HS_k(J(G_k)) is generated by the product of all cycle vertices times the two monomials y_1y_3...y_{2k-1} and y_2y_4...y_{2k}. This ideal is generated in degree 3k, yet its Castelnuovo-Mumford regularity is 4k-1 by a lemma of Woodroofe, so it cannot have a linear resolution. Consequently HS_k(J(G_k)) does not have linear quotients, directly contradicting the assertions in [7, Theorem 4.1, 4.2, Corollary 4.11], [8, Theorem 4.8], and [7, Conjecture 4.4]. On the positive side, the paper proves that for every Cohen-Macaulay chordal graph G, HS_k(J(G)) has linear quotients for all k, and for every Cohen-Macaulay Cameron-Walker graph G, HS_k(J(G)) is weakly polymatroidal; the same holds for clique corona graphs Γ◦H in which every clique K_{t_i} has t_i≥2.","pith_inferences":["One can test the boundary of the positive result for clique corona graphs: the counterexample uses t_i=1, and the paper proves weakly polymatroidal for all t_i≥2; intermediate cases with mixed t_i=1 and t_i≥2 remain untested and may still fail or only satisfy linear quotients without weak polymatroidality.","The regularity gap (4k-1 vs 3k) grows with k, so the failure of linear resolution is not an isolated low-degree artifact; larger k gives increasingly non-linear behavior.","The special ordering (∗) used for chordal graphs might be adapted to other clique-whiskered classes whose underlying graph is chordal-like, such as forests or block graphs, to test whether linear quotients persist.","Since the contradiction depends on G_k being very well-covered, a careful check of the very well-covered property would pinpoint which hypothesis in [7, Theorem 4.1] fails; if G_k is indeed very well-covered, the error must lie in the proof of that theorem."],"forward_implications":["The previously published theorems [7, Theorem 4.1, 4.2, Corollary 4.11], [8, Theorem 4.8] and the conjecture [7, Conjecture 4.4] are false; any proof of them must contain an error.","The class of Cohen-Macaulay very well-covered graphs is too broad for the linear quotient property of homological shift ideals; the property holds only for specific subclasses.","For Cohen-Macaulay chordal graphs, the special ordering on minimal generators gives linear quotients of HS_k(J(G)) for all k, providing a template for future proofs of linear quotients.","For Cohen-Macaulay Cameron-Walker graphs and for clique corona graphs whose cliques each have size at least two, the homological shift ideals are weakly polymatroidal, hence have linear quotients.","The counterexample graphs G_k are clique corona graphs with t_i=1, so the condition t_i≥2 in the positive clique-corona theorem is exactly what separates the two behaviors."],"supporting_citations":[{"why":"The theorem 4.1, 4.2, and Corollary 4.11 that the counterexample contradicts; the baseline assertion that all homological shift ideals of vertex cover ideals of Cohen-Macaulay very well-covered graphs have linear quotients.","marker":"[7]"},{"why":"Theorem 4.8, which asserts linear quotients for all homological shift ideals of whiskered graphs; contradicted by the same construction.","marker":"[8]"},{"why":"Supplies the clique-whiskered graph construction and the fact that these graphs are Cohen-Macaulay; G_k is a clique-whiskered graph.","marker":"[6]"},{"why":"Provides the definition of homological shift ideals and the fact used in Remark 2.6 that the minimal generators of HS_k are x^C x^σ with σ of size k.","marker":"[18]"},{"why":"Lemma 8 gives the regularity 4k-1 of the specific non-linear ideal, the numerical fact that drives the contradiction.","marker":"[30]"},{"why":"Standard facts used to conclude that an equigenerated ideal is linear iff its regularity equals its degree, and that linear quotients imply linear resolution.","marker":"[16]"}],"fun_headline_variants":["Whiskered cycles smash shift-ideal conjecture","Shift ideals fail for whiskered even cycles","No linear resolution for shift ideals here","Even-cycle whiskers break homological shifts","Counterexample: whiskered graphs foil shift ideals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The contradiction to the earlier theorem relies on the graph G_k being very well-covered, a property the paper asserts in the abstract but does not prove in the body; if G_k were not very well-covered, the contradiction would not apply to that theorem.","fun_headline_variants_meta":{"raw":{"variants":["Whiskered cycles smash shift-ideal conjecture","Shift ideals fail for whiskered even cycles","No linear resolution for shift ideals here","Even-cycle whiskers break homological shifts","Counterexample: whiskered graphs foil shift ideals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000232,"raw_usage":{"total_tokens":1572,"prompt_tokens":1112,"completion_tokens":460,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":728,"completion_tokens_details":{"reasoning_tokens":393}},"tokens_in":728,"tokens_out":460,"duration_ms":5044,"temperature":1.0,"reasoning_tokens":393,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:35:39.788834+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for k=2, the minimal free resolution of the ideal (x_1x_2x_3x_4)⟨y_1y_3, y_2y_4⟩ in K[x_1,...,x_4,y_1,...,y_4] using any computer algebra system; if the Castelnuovo-Mumford regularity is 6 (the generation degree) rather than 7, then the ideal would have a linear resolution and the counterexample would fail. Alternatively, verify the very well-covered property of the whiskered 4-cycle by checking |V|=2α(G) and that every maximal independent set has size α.","supporting_citations":[{"cited_title":"Crupi and A","cited_arxiv_id":null,"evidence_quote":"The theorem 4.1, 4.2, and Corollary 4.11 that the counterexample contradicts; the baseline assertion that all homological shift ideals of vertex cover ideals of Cohen-Macaulay very well-covered graphs have linear quotients."},{"cited_title":"Crupi and A","cited_arxiv_id":null,"evidence_quote":"Theorem 4.8, which asserts linear quotients for all homological shift ideals of whiskered graphs; contradicted by the same construction."},{"cited_title":"Cook, II and U","cited_arxiv_id":null,"evidence_quote":"Supplies the clique-whiskered graph construction and the fact that these graphs are Cohen-Macaulay; G_k is a clique-whiskered graph."},{"cited_title":"Herzog, S","cited_arxiv_id":null,"evidence_quote":"Provides the definition of homological shift ideals and the fact used in Remark 2.6 that the minimal generators of HS_k are x^C x^σ with σ of size k."},{"cited_title":"Woodroofe","cited_arxiv_id":null,"evidence_quote":"Lemma 8 gives the regularity 4k-1 of the specific non-linear ideal, the numerical fact that drives the contradiction."},{"cited_title":"Herzog and T","cited_arxiv_id":null,"evidence_quote":"Standard facts used to conclude that an equigenerated ideal is linear iff its regularity equals its degree, and that linear quotients imply linear resolution."}],"review_version":1}