Powers of Edge Ideals with Linear Quotients

Etan Basser; Mario Stinson-Maas; Rachel Diethorn; Robert Miranda

Powers of Edge Ideals with Linear Quotients

T0 review · 0 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proves that for every anticycle graph on five or more vertices, every power of its edge ideal beyond the first admits an explicit ordering with linear quotients, even though the edge ideal itself does not.

desk verdict Solid, publishable paper: explicit linear quotient ordering for powers of anticycle edge ideals resolves Hoefel–Whieldon, plus a clean constructive proof for all quadratic monomial ideals with linear quotients; the main proof is long but sound, with one external dependency worth an independent check. read the letter →

arxiv 2412.03468 v3 pith:24NJPENO submitted 2024-12-04 math.AC

Etan Basser , Rachel Diethorn , Robert Miranda , Mario Stinson-Maas This is my paper · ORCID

classification math.AC MSC 13D0213F5513P2005E40

keywords edgeidealslinearquotientspowersofanticyclegraphsgap-freewhiskerBettinumbersprojectivedimension

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

An ideal with linear quotients can be resolved with linear maps, and its Betti numbers can be read off from the sizes of certain quotients. This paper asks which powers of edge ideals inherit that property. The central answer is that the edge ideal of an anticycle graph—the complement of a cycle on at least five vertices—does not have linear quotients, but its second and every higher power do, via an explicit and simple generator ordering. That settles an open question and supplies the first family of gap-free graphs whose edge ideals behave this way. The same paper gives a general construction showing that whenever a quadratic monomial ideal has linear quotients, all of its powers do, and it derives closed formulas for projective dimension and Betti numbers of powers of whisker graphs.

What carries the argument

The machinery is an explicit ordering of minimal generators, verified by a variable-divisibility criterion. Lemma 2.3 says an ordering has linear quotients exactly when, for every earlier/later pair $(M_1,M_2)$, some earlier generator $M_3$ has $M_3/\gcd(M_3,M_2)$ a variable dividing $M_1/\gcd(M_1,M_2)$. Against that criterion, the paper sets Construction 3.3 for quadratic ideals—formal products of $k$ edges ordered by reverse lexicographic order on their exponent vectors, with duplicates removed—and Construction 5.1 for anticycle powers, a two-block ordering whose second block reuses the antipath power ordering and whose one transposition (moving $(x_1x_{n-1})^k$ after $D$) repairs the only failure of the naive concatenation. Lemma 3.10 partitions all pairs of generators into three cases, and Theorem 5.5's proof checks each case to show every quotient is variable-generated.

What would settle it

For a fixed $n\ge5$ and $k\ge2$, compute the ideals $((\text{earlier generators}):(\text{current generator}))$ for the ordering of Construction 5.1 and inspect their minimal generators. The theorem is false as soon as any quotient contains a minimal generator that is not a single variable; an independent finite check for $A_6^3$ or $A_7^2$ would settle it. Equivalently, one pair $M_1$ preceding $M_2$ with no $M_3$ satisfying the divisibility condition of Lemma 2.3 disproves the claim.

Watch

Extended reading notes

Core claim

The paper proves two theorems. Theorem 5.5: for $n\ge5$ and $k\ge2$, ordering the minimal generators of $I(A_n)^k$ by Construction 5.1—first the generators divisible by $x_n$ in lex order with $x_n>x_2>\cdots>x_{n-1}>x_1$, then the rest in lex order $x_1>x_2>\cdots>x_{n-1}$, with $(x_1x_{n-1})^k$ moved directly after the distinguished generator $D=(x_1x_{n-1})^{k-1}(x_2x_{n-1})$—yields a linear quotient ordering. Because $I(A_n)$ itself is known not to have linear quotients, this answers the open question in the affirmative and provides the first gap-free graphs whose edge ideal powers have linear quotients while the ideal itself does not. Theorem 3.11: if a quadratic monomial ideal $I(G)$ has a linear quotient ordering, then the reverse-lexicographic ordering of the formal edge products $m_1^{\alpha_1}\cdots m_r^{\alpha_r}$, with repeated vertex monomials deleted, is a linear quotient ordering of $I(G)^k$ for every $k$. As a consequence the paper computes explicit projective dimension and Betti numbers for powers of whisker graph edge ideals.

Load-bearing premise

The main theorem inherits the previously proved fact that every power of the antipath's edge ideal has linear quotients in lexicographic order; if that cited fact is wrong or misquoted, the verification of the second block of the anticycle ordering collapses.

Editorial extensions

If this is right

For every $n\ge5$ and $k\ge2$, the anticycle power $I(A_n)^k$ has a linear resolution, not merely a linear quotient ordering, because linear quotients feed the mapping-cone resolution.
Anticycles become the first known gap-free graphs whose edge ideals fail linear quotients while all second and higher powers have them, making them a test case for the conjecture that high powers of gap-free edge ideals have linear resolutions.
Any quadratic monomial ideal that admits linear quotients has all of its powers admitting linear quotients, with an explicit ordering that depends only on the original ordering; this makes the Betti numbers of such powers computable from quotient counts.
For whisker graphs $W_{r,\ell}$, the projective dimension of $I(W_{r,\ell})^k$ is $r+\ell$ for $k\ge2$ and $\max(r,\ell)$ for $k=1$, with closed binomial formulas for all Betti numbers.
The paper's Example 3.5 gives a quadratic ideal whose square's lexicographic ordering fails the linear quotient test, disproving Conjecture 4.1 of a related preprint about efficient orderings and powers.

Reading between the lines

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

The block-ordering strategy may extend to other graphs whose edge ideals are nearly antipath-like: the only place the anticycle combinatorics enters is the verification that the two blocks $F$ and $S$ interact, so any graph whose ideal quotients match those blocks would inherit the theorem.
The revlex formal-edge ordering in Theorem 3.11 is algorithmic; applying it to other tree or chordal families would yield projective dimension and Betti number formulas analogous to the whisker ones, simply by counting quotient sizes.
The counterexample to Conjecture 4.1 suggests that the efficient-ordering notion in that conjecture is too rigid; a quotient-aware invariant, rather than the ordering alone, may be the right object to track across powers.
Since an anticycle edge ideal is far from linear quotients while its powers are not, failure of linear quotients in an edge ideal may be a poor predictor of failure in its powers, so conjectures phrased only in terms of the ideal itself may need reformulation in terms of the powers.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I read this carefully and agree with the ACCEPT verdict. The main news is Theorem 5.5: the second and higher powers of anticycle edge ideals admit linear quotient orderings even though the edge ideals themselves do not. That answers an open question of Hoefel and Whieldon and gives the first gap-free graphs whose edge ideals have this property on all powers. The ordering in Construction 5.1 is explicit and surprisingly simple, and the Macaulay2 verification for A6^2 is a useful independent check.

The paper also does something genuinely useful in Section 3. Theorem 3.11 constructs an explicit revlex ordering on edge decompositions proving that powers of any quadratic monomial ideal with linear quotients again have linear quotients. This was known existentially, but the previous arguments went through the Rees algebra or the x-condition; here the ordering depends only on the original linear quotient ordering. Example 3.5 is a valid counterexample to [12, Conjecture 4.1] under the stated lex convention, and the whisker graph Betti formulas in Theorem 4.5 are new and follow cleanly from Lemma 4.4.

The soft spot is the proof of Theorem 5.5 itself. It is a long case analysis, and the exhaustiveness is asserted rather than transparently organized. I checked the case split in Case 3: after fixing the first differing exponent in the lex order, Subcases 3.1–3.4 do cover every possibility, and the degree-counting contradictions are sound. The genuinely exposed point is Case 2, which relies on [21, Proposition 3.1] for the antipath powers: that the lex order x1 > x2 > ... > x_{n-1} is a linear quotient ordering of I(P_{n-1})^k for every k. That proposition is not proved in this paper, and the Macaulay2 check only covers n=6, k=2. If it were misquoted or had a hidden k restriction, Theorem 5.5 would need repair. But citing a prior result is standard, and I see no sign it is false or misapplied. So this is a 'verify at referee time' item, not a defect.

The computational package (isLinear, getQuotients, findLinearOrderings) with code on GitHub is a real asset for others working on this problem. The paper is aimed at commutative algebraists studying edge ideals, linear quotients, and resolutions of powers; it gives constructive tools and a clean new example. I would bring it to a reading group and would cite it.

Recommendation: send to a serious referee. If I were the editor, I would ask the referee to check the invocation of [21, Proposition 3.1] and spot-check Subcase 3.3, but I expect the paper to pass. It is a solid advance in a specialized area, not a foundational breakthrough, and it deserves publication in a good commutative algebra journal.

Referee Report

0 major / 4 minor

Summary. The paper studies linear quotient orderings for powers of edge ideals. It constructs, in Construction 3.3, a revlex ordering of edge decompositions and proves in Theorem 3.11 that every power of a quadratic monomial ideal with linear quotients again admits linear quotients; this gives a constructive proof of a known result of Herzog, Hibi, and Zheng. In Section 4 the ordering is applied to whisker graphs, yielding explicit projective dimension and Betti number formulas in Theorem 4.5. The main result is Theorem 5.5, which gives an explicit ordering O_n^(k) showing that all powers I(A_n)^k with n ≥ 5 and k ≥ 2 admit linear quotients, although I(A_n) itself does not; this answers Hoefel and Whieldon's Question 5.1. Example 3.5 gives a counterexample to a conjecture of Erey et al. Section 6 provides Macaulay2 methods and verifications for the constructions.

Significance. If the results are correct, Theorem 5.5 resolves an open question in the affirmative and supplies the first family of gap-free graphs whose edge ideals fail to have linear quotients while all higher powers do. Theorem 3.11 is a useful constructive strengthening of the Herzog-Hibi-Zheng theorem, and the whisker graph formulas in Theorem 4.5 are explicit and checkable. The paper is notable for giving concrete, reproducible computational support: the Macaulay2 code is available in [30], and Example 6.1 verifies the anticycle ordering for A_6^2. I read the proofs as internally consistent: the case analysis in Theorem 5.5 is lengthy but the subcases are exhaustive as presented, and the counterexample in Example 3.5 is valid under the stated lex convention. The only external input that is load-bearing for the main theorem is the cited [21, Proposition 3.1]; I found no evidence that it is misquoted or inapplicable.

minor comments (4)

[§5, Theorem 5.5, Case 2] The proof of Case 2 depends on the cited [21, Proposition 3.1] for the statement that the lex ordering x_1 > x_2 > ... > x_{n-1} gives a linear quotient ordering of I(P_{n-1})^k for every k. Since this is the single most delicate external dependency and the included Macaulay2 check covers only n = 6, k = 2, the authors should state the proposition explicitly, with the exact hypotheses on n and k, to make the dependency transparent.
[§4, Theorem 4.5] The statement of Theorem 4.5 does not specify the allowed ranges of r and ℓ. If r = ℓ = 0 is allowed, the graph W_{0,0} is a single edge and the claimed formula gives pd = 0 for k ≥ 2 instead of the correct value 1; the authors should add the standing assumption r + ℓ ≥ 1 (or r, ℓ ≥ 1, as intended).
[§5, opening paragraph] The notation I(P_n) for the antipath is nonstandard and can be confused with the edge ideal of an ordinary path. The authors should use a clearer notation such as I(\overline{P_n}) or explicitly remind the reader at every occurrence that P_n denotes the antipath.
[§6, Example 6.1] The Macaulay2 output of getQuotients is difficult to read in print because of the line breaks and spacing; reformatting the output as a vertical list or table would improve readability and make the verification easier to follow.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity identified: the proofs are direct constructions from explicit orderings and external lemmas, with no fitted parameters or self-citation chain bearing the central claims.

full rationale

The central claims are derived directly from explicit constructions rather than from the conclusions they are meant to prove. Theorem 3.11 assumes only that the quadratic monomial ideal I(G) admits linear quotients and then proves, by induction and the technical lemmas in Section 3, that the revlex edge-decomposition list R(G)^k for I(G)^k is a linear quotient ordering up to repetition; no quantity is fitted to the target result. Theorem 5.5 constructs the ordering O_n^{(k)} in Construction 5.1 and verifies it against the criterion in Lemma 2.3 through a case analysis on pairs of generators. The only exposed external dependency is the citation of [21, Proposition 3.1] in Case 2, which supplies the linear quotient property for the lexicographic ordering on I(P_{n-1})^k for the antipath sub-ordering S. That citation is to prior work by Hoefel and Whieldon, not by the present authors; it concerns the related antipath edge ideal rather than the target anticycle ideal, so it is not an importation of the paper's own conclusion. The Macaulay2 computations in Section 6 are described as checks and examples, not as inputs to the proof, and the self-citation to the authors' own code repository [30] is not load-bearing for any mathematical claim. The derivation chain is therefore self-contained with respect to circularity: no step reduces by construction to its own input, and no fitted parameter is renamed as a prediction. The cited antipath proposition deserves independent verification, but that is a correctness or completeness concern, not a circularity concern.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard lemmas from monomial ideal theory, one cited result about antipath powers, and the graph-theoretic identification of the anticycle with the antipath on fewer vertices. No free parameters or invented entities are introduced.

assumptions (5)

standard math Lemma 2.3: an ordering yields linear quotients iff for every pair m_j, m_i with j < i, there is m_h with h < i and a variable x such that m_h/gcd(m_h,m_i) = x and x divides m_j/gcd(m_j,m_i).
Used throughout the paper to verify linear quotient orderings; cited from [18, Lemma 8.2.3].
standard math Proposition 2.4: for an ideal with linear quotients, the projective dimension is the maximum number of generators of the ideal quotients, and Betti numbers are sums of binomial coefficients.
Used in Section 4 to compute projective dimension and Betti numbers; cited from [29], [20], and [18, Corollary 8.2.2].
standard math Herzog-Hibi-Zheng theorem: for a quadratic monomial ideal, linear resolution, linear quotients, and linear resolutions of all powers are equivalent.
Motivates the problem and is the background for the constructive result; cited from [19, Theorem 3.2].
domain assumption The lexicographic ordering x1 > x2 > ... > x_{n-1} is a linear quotient ordering of I(P_{n-1})^k for the antipath P_{n-1}.
Used in Case 2 of Theorem 5.5 for the S sub-ordering. This is a cited result from [21, Proposition 3.1] rather than proved in the paper.
domain assumption The restriction of the anticycle A_n to vertices x1 through x_{n-1} is the antipath P_{n-1}.
Used in Case 2 of Theorem 5.5 to identify the S generators not divisible by x_n with minimal generators of I(P_{n-1})^k.

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Pith. "Pith review of Powers of Edge Ideals with Linear Quotients." pith.science (2026). https://pith.science/paper/24NJPENO

@misc{pith2026241203468,
  author       = {Pith},
  title        = {Pith review of: Powers of Edge Ideals with Linear Quotients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24NJPENO}},
  note         = {Machine review of arXiv:2412.03468}
}

read the original abstract

We prove that second and higher powers of the edge ideals of anticycles admit linear quotient orderings, although the edge ideals themselves do not, thus resolving an open question of Hoefel and Whieldon in the affirmative and providing the first class of gap-free graphs whose edge ideals satisfy this property on their powers. We also construct an explicit and straightforward linear quotient ordering for any power of a quadratic monomial ideal which admits linear quotients. This expands on a well-known result of Herzog, Hibi, and Zheng. As a consequence, we give explicit formulas for the projective dimension and Betti numbers of the edge ideals of whisker graphs.

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Forward citations

Cited by 2 Pith papers

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Works this paper leans on

30 extracted references · 29 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [30]

Stinson-Maas, Linear Quotients

M. Stinson-Maas, Linear Quotients. https://github.com/Mario730/linearquotients, July 2024. 24 E. BASSER, R. DIETHORN, R. MIRANDA, AND M. STINSON-MAAS Department of Mathematics, Yale University, New Haven, CT Email address : etan.basser@yale.edu Department of Mathematics, Oberlin College, Oberlin, OH Email address : rdiethor@oberlin.edu Department of Mathe...

work page 2024

[2] [1]

Banerjee, The regularity of powers of edge ideals , J

A. Banerjee, The regularity of powers of edge ideals , J. Algebraic Combin., 41 (2015), pp. 303–321

work page 2015

[3] [2]

Banerjee, S

A. Banerjee, S. K. Beyarslan, and H. T. H `a, Regularity of powers of edge ideals: from local properties to global bounds, Algebr. Comb., 3 (2020), pp. 839–854

work page 2020

[4] [3]

Beyarslan, H

S. Beyarslan, H. T. H `a, and T. N. Trung , Regularity of powers of forests and cycles , J. Algebraic Combin., 42 (2015), pp. 1077–1095

work page 2015

[5] [4]

Bigdeli , Edge ideals with almost maximal finite index and their powers , J

M. Bigdeli , Edge ideals with almost maximal finite index and their powers , J. Algebraic Combin., 54 (2021), pp. 947–978

work page 2021

[6] [5]

Bigdeli, J

M. Bigdeli, J. Herzog, and R. Zaare-Nahandi , On the index of powers of edge ideals , Comm. Algebra, 46 (2018), pp. 1080–1095

work page 2018

[7] [6]

Conca and J

A. Conca and J. Herzog , Castelnuovo-Mumford regularity of products of ideals , Collect. Math., 54 (2003), pp. 137–152

work page 2003

[8] [7]

S. M. Cooper, S. El Khoury, S. F aridi, S. Mayes-Tang, S. Morey, L. M. ¸Sega, and S. Spiroff, Simplicial resolutions of powers of square-free monomial ideals , Algebr. Comb., 7 (2024), pp. 77–107

work page 2024