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On the homological shifts of cover ideals of Cohen-Macaulay graphs

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Pith's one-line read 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…

desk verdict The counterexample is real and important; the chordal/Cameron–Walker proofs are solid, but the clique corona section and internal references need work. read the letter →

arxiv 2506.01810 v1 pith:3SHM3DYU submitted 2025-06-02 math.AC math.CO

Amit Roy , Kamalesh Saha This is my paper
classification math.ACmath.CO MSC 13D0205E4013F5513H10
keywords homologicalshiftidealslinearquotientsvertexcoverCohen-MacaulaygraphschordalCameron-Walkerclique-whiskeredweaklypolymatroidal
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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).

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 (2)
  1. [Section 2, proof of Theorem 2.14] 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).
  2. [Section 2, Theorem 2.17] 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.
minor comments (4)
  1. [Abstract and Section 2 (Theorem 2.7)] 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.
  2. [Throughout Section 2] 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.
  3. [Section 2, proof of Theorem 2.7] 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.
  4. [Section 2, Theorem 2.17 statement] The statement 'and thus, linear quotients for all k≥0' should read 'and thus, has linear quotients for all k≥0'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the counterexample is derived from explicit generators and an independent regularity computation.

full rationale

The central claim (Theorem 2.7) is derived by explicitly computing the minimal generators of HS_k(J(G_k)) via the structural description of homological shift ideals of clique-whiskered graphs (Proposition 2.4 and Remark 2.6, citing Herzog et al. [18]), then computing the regularity directly: the ideal equals (product of all x_i) times the ideal generated by the two monomials of odd-indexed y's and even-indexed y's, is generated in degree 3k, and the lcm of its two generators has degree 4k, so its regularity is 4k-1. The regularity statement cites Woodroofe [30, Lemma 8], an external result, and the criterion reg = d for equigenerated ideals with linear resolutions is from Herzog and Hibi [16]. No parameter is fitted and no target result is assumed. The claimed contradiction with Crupi and Ficarra [7, Theorem 4.1] is a benchmark against an external theorem, not an input to the derivation. The only weakness, the unproved assertion that G_k is very well-covered, is not circular: it is a direct property of whiskered graphs on bipartite 2k-cycles, and no part of the proof of Theorem 2.7 depends on it. The positive results (Theorems 2.11, 2.14, and 2.17) use Betti splitting, weak polymatroidality, and classification theorems from the literature as tools rather than as conclusions. Self-citations by the authors are absent; the citations to [18], [30], [16], [6], [20], and [17] are independent supporting results.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No fitted parameters or new entities are introduced. The graph G_k has a parameter k, which is a theorem index rather than a fitted value. The proof rests on standard results in combinatorial commutative algebra, listed as axioms.

assumptions (7)
  • domain assumption Minimal generators of HS_k(I) for an ideal I with linear quotients are m_i x^σ with σ a subset of the colon set of m_i of size k.
    Used to compute HS_k(J(G_k)) and repeated in proofs; cited from Herzog et al. [18, Page 4].
  • domain assumption Every clique-whiskered graph is vertex decomposable and hence Cohen-Macaulay.
    Used to conclude G_k and other graphs are Cohen-Macaulay; cited from Cook and Nagel [6].
  • domain assumption Every whisker graph W(H) is very well-covered.
    Needed for the contradiction with [7, Theorem 4.1]; stated in the abstract but not proved in the body.
  • domain assumption A chordal graph is Cohen-Macaulay if and only if it is a clique-whiskered graph.
    Transfers the study of chordal graphs to the clique-whiskered setting; cited from Herzog-Hibi-Zheng [17].
  • domain assumption Woodroofe's Lemma [30, Lemma 8] gives the regularity used to show HS_k(J(G_k)) is not linear.
    External result used to compute reg(HS_k(J(G_k))) = 4k-1.
  • standard math An equigenerated ideal has a linear resolution if and only if its regularity equals the common degree of its generators.
    Standard fact used in the proof of Theorem 2.7.
  • domain assumption Every weakly polymatroidal ideal has linear quotients.
    Used to conclude the linear quotient property for Cameron-Walker and clique corona graphs; cited from Mohammadi-Moradi [25].

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Pith. "Pith review of On the homological shifts of cover ideals of Cohen-Macaulay graphs." pith.science (2026). https://pith.science/paper/3SHM3DYU

@misc{pith2026250601810,
  author       = {Pith},
  title        = {Pith review of: On the homological shifts of cover ideals of Cohen-Macaulay graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SHM3DYU}},
  note         = {Machine review of arXiv:2506.01810}
}
abstract

For a non-negative integer $k$, let $\mathrm{HS}_{k}(J(G))$ denote the $k^{\text{th}}$ homological shift ideal of the vertex cover ideal $J(G)$ of a graph $G$. For each $k\geq 2$, we construct a Cohen-Macaulay very well-covered graph $G_k$ which is both Cohen-Macaulay bipartite and a whiskered graph so that $\mathrm{HS}_{k}(J(G))$ does not have a linear resolution. This contradicts several results as well as disproves a conjecture in [J. Algebra, $\mathbf{629}$, (2023), 76-108] and [Mediterr. J. Math., $\mathbf{21}$, 135 (2024)]. The graphs $G_k$ are also examples of clique-whiskered graphs introduced by Cook and Nagel, which include Cohen-Macaulay chordal graphs, Cohen-Macaulay Cameron-Walker graphs, and clique corona graphs. Surprisingly, for Cohen-Macaulay chordal graphs, we can use a special ordering on the minimal generators to show that $\mathrm{HS}_{k}(J(G))$ has linear quotients for all $k$. Moreover, for all Cohen-Macaulay Cameron-Walker graphs and certain clique corona graphs, we show that $\mathrm{HS}_{k}(J(G))$ is weakly polymatroidal, and thus, has linear quotients for all $k$.

Figures

Figures reproduced from arXiv: 2506.01810 by the authors.

Figure 1
Figure 1. The graph Gk = W(C2k). It is easy to see that Gk is an example of a clique-whiskered graph. In fact, Gk is the whiskered graph on the cycle of length 2k. Indeed, if C2k denotes the cycle of length 2k on the vertex set {x1, . . . , x2k} with clique vertex-partition π = {{x1}, . . . , {x2k}}, then Gk = C π 2k . Using the description of the minimal generators of HSk(J(Gπ )) from Proposition 2.4 and Proposition 2.6, we … view at source ↗

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Works this paper leans on

30 extracted references · 29 canonical work pages

  1. [1]

    S. Bandari. Polymatroidal ideals and linear resolution.J. Algebr. Syst., 11(2):147–153, 1, 2024. 2, 15

    work page 2024

  2. [2]

    S. Bayati. Multigraded shifts of matroidal ideals.Arch. Math. (Basel), 111(3):239–246, 2018. 2, 15

    work page 2018

  3. [3]

    S. Bayati. A quasi-additive property of homological shift ideals.Bull. Malays. Math. Sci. Soc., 46(3):Paper No. 111, 17, 2023. 2

    work page 2023

  4. [4]

    Bayati, I

    S. Bayati, I. Jahani, and N. Taghipour. Linear quotients and multigraded shifts of Borel ideals.Bull. Aust. Math. Soc., 100(1):48–57, 2019. 2

    work page 2019

  5. [5]

    T. Chau, K. K. Das, and A. Maithani. Edge ideals with linear quotients and without homological linear quotients.arXiv preprint, arXiv:2503.11424, 2025. 2

    work page arXiv 2025

  6. [6]

    Cook, II and U

    D. Cook, II and U. Nagel. Cohen-Macaulay graphs and face vectors of flag complexes.SIAM J. Discrete Math., 26(1):89–101, 2012. 3, 4, 5, 6, 8

    work page 2012

  7. [7]

    Crupi and A

    M. Crupi and A. Ficarra. Very well-covered graphs by Betti splittings.J. Algebra, 629:76–108, 2023. 2, 3, 8

    work page 2023

  8. [8]

    Crupi and A

    M. Crupi and A. Ficarra. Very well-covered graphs via the Rees algebra.Mediterr. J. Math., 21(4):Pa- per No. 135, 17, 2024. 2, 3

    work page 2024

Show all 30 references
  1. [9]

    T. H. Do and M. H. Pham. The size of Betti tables of edge ideals of clique corona graphs.Arch. Math. (Basel), 118(6):577–586, 2022. 4

    2022

  2. [10]

    J. A. Eagon and V. Reiner. Resolutions of Stanley-Reisner rings and Alexander duality.J. Pure Appl. Algebra, 130(3):265–275, 1998. 2

    1998

  3. [11]

    A. Ficarra. Homological shifts of polymatroidal ideals.To appear in the Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie, arXiv:2205.04163, 2022. 2, 15

    2022 arXiv

  4. [12]

    Ficarra and J

    A. Ficarra and J. Herzog. Dirac’s theorem and multigraded syzygies.Mediterr. J. Math., 20(3):Paper No. 134, 18, 2023. 2

    2023

  5. [13]

    Ficarra and A

    A. Ficarra and A. A. Qureshi. The homological shift algebra of a monomial ideal.arXiv:2412.21031,

    arXiv

  6. [14]

    Ficarra and A

    A. Ficarra and A. A. Qureshi. Edge ideals and their asymptotic syzygies.arXiv:2501.07319, 2025. 2

    2025 arXiv

  7. [15]

    C. A. Francisco, H. T. H` a, and A. Van Tuyl. Splittings of monomial ideals.Proc. Amer. Math. Soc., 137(10):3271–3282, 2009. 8

    2009

  8. [16]

    Herzog and T

    J. Herzog and T. Hibi.Monomial ideals, volume 260 ofGraduate Texts in Mathematics. Springer- Verlag London, Ltd., London, 2011. 2, 3, 8

    2011

  9. [17]

    Herzog, T

    J. Herzog, T. Hibi, and X. Zheng. Cohen-Macaulay chordal graphs.J. Combin. Theory Ser. A, 113(5):911–916, 2006. 9

    2006

  10. [18]

    Herzog, S

    J. Herzog, S. Moradi, M. Rahimbeigi, and G. Zhu. Homological shift ideals.Collectanea Mathematica, 72:157–174, 2021. 2, 7, 15

    2021

  11. [19]

    Herzog, S

    J. Herzog, S. Moradi, M. Rahimbeigi, and G. Zhu. Some homological properties of Borel type ideals. Comm. Algebra, 51(4):1517–1531, 2023. 2

    2023

  12. [20]

    T. Hibi, A. Higashitani, K. Kimura, and A. B. O’Keefe. Algebraic study on Cameron-Walker graphs. J. Algebra, 422:257–269, 2015. 3, 13 HOMOLOGICAL SHIFTS OF COVER IDEALS 17

    2015

  13. [21]

    Kimura, M

    K. Kimura, M. R. Pournaki, S. A. Seyed Fakhari, N. Terai, and S. Yassemi. A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra.Res. Math. Sci., 9(2):Paper No. 29, 18, 2022. 3

    2022

  14. [22]

    Kokubo and T

    M. Kokubo and T. Hibi. Weakly polymatroidal ideals.Algebra Colloq., 13(4):711–720, 2006. 4, 12

    2006

  15. [23]

    Lu and Z

    D. Lu and Z. Wang. On powers of cover ideals of graphs.Osaka J. Math., 61(2):247–259, 2024. 15

    2024

  16. [24]

    Miller and B

    E. Miller and B. Sturmfels.Combinatorial commutative algebra, volume 227 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 2005. 1

    2005

  17. [25]

    Mohammadi and S

    F. Mohammadi and S. Moradi. Weakly polymatroidal ideals with applications to vertex cover ideals. Osaka J. Math., 47(3):627–636, 2010. 4, 12

    2010

  18. [26]

    Moradi and F

    S. Moradi and F. Khosh-Ahang. On vertex decomposable simplicial complexes and their Alexander duals.Math. Scand., 118(1):43–56, 2016. 8

    2016

  19. [27]

    Muta and N

    Y. Muta and N. Terai. On minimal free resolutions of the cover ideals of clique-whiskered graphs. arXiv:2505.04248, 2025. 3

    2025

  20. [28]

    Taghipour, S

    N. Taghipour, S. Bayati, and F. Rahmati. Homological linear quotients and edge ideals of graphs. Bull. Aust. Math. Soc., 110(2):291–302, 2024. 2

    2024

  21. [29]

    R. H. Villarreal. Cohen-Macaulay graphs.Manuscripta Math., 66(3):277–293, 1990. 3

    1990

  22. [30]

    Woodroofe

    R. Woodroofe. Matchings, coverings, and Castelnuovo-Mumford regularity.J. Commut. Algebra, 6(2):287–304, 2014. 8 Chennai Mathematical Institute, India Email address:amitiisermohali493@gmail.com Chennai Mathematical Institute, India Email address:kamalesh.saha44@gmail.com; ksah...

    2014

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