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Gapfree graphs and powers of edge ideals with linear quotients

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.06467 v2 pith:IIXOVXKC submitted 2024-12-09 math.AC math.CO

Nursel Erey , Sara Faridi , Tài Huy Hà , Takayuki Hibi , Selvi Kara , Susan Morey This is my paper · ORCID
classification math.ACmath.CO MSC 05E4013D02
keywords edgeidealgapfreegraphlinearquotientsresolutionpowersofidealsCDCCvertexduplicationadmissibleordering
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 6 minor

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).

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 (3)
  1. [Proposition 5.4] 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.
  2. [Theorem 6.4] 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.
  3. [Proposition 2.3] 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.
minor comments (6)
  1. [Abstract and Section 6] 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.
  2. [Proposition 5.2] 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}.
  3. [Proposition 5.4] In the displayed order for I(G)^2, the term 'e3e9 = (ax)qz)' is missing a parenthesis; it should read 'e3e9 = (ax)(qz)'.
  4. [Proposition 4.2, Case 1] 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.
  5. [Section 5, first paragraph] 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.
  6. [Theorem 5.7] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain is a proof of conditional implications; the finite verification gaps in Propositions 5.2/5.4 are missing certificates, not circular reductions.

full rationale

The paper's central claims are proved by induction and cited external theorems, and no step derives a conclusion from an assumption equivalent to it. Proposition 2.3 proves preservation of linear quotients under duplication with a full argument; Theorem 3.4 gives an if-and-only-if for expansion with a complete proof. Propositions 4.2, 5.2, and 5.4 use induction from a verified base (the base for I(G)^2 is claimed by direct computation rather than exhibited), and the induction step proves that linear quotients for s-1 imply linear quotients for s under the efficient ordering. This is an implication, not a tautology. Theorem 5.7 then applies duplication and the proven Proposition 2.3 and Proposition 5.4. The open conjectures (1.1, 1.2, 4.1) are explicitly stated as conjectures and are not used as inputs. Known results such as Froberg's theorem, Nevo-Peeva's negative example, and Herzog-Hibi-Zheng are external standard results, not self-citations that carry the argument. The paper honestly notes limitations in Remarks 3.6, 6.7, and 6.8. The only weaknesses are the unshown finite verifications in Propositions 5.2 and 5.4 ("It is now a direct computation to verify..." and "One can verify..."), which are gaps in the exhibited proof of the base case, not circular reasoning: they do not assume the desired conclusion nor fit an output to an input. A missing finite check is a correctness risk, but the prompt restricts circularity findings to exhibited reductions to inputs, and none is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claims rest only on standard algebraic facts, graph-theoretic definitions, and finite computational assertions that are listed but not machine-checked. There are no fitted parameters or invented entities.

assumptions (3)
  • standard math A monomial ideal generated in a single degree with linear quotients has linear resolution (Herzog-Hibi, [13, Proposition 8.2.1]).
    Bridges linear quotients to linear resolution; used in the introduction and throughout.
  • domain assumption If I(G)^q has linear resolution for some q, then G is gapfree (Francisco-Ha-Van Tuyl, cited via [18]).
    Used to justify necessity in Theorem 3.4 and in the introduction's framing of Conjecture 1.2.
  • standard math Froberg's theorem: I(G) has linear resolution iff the complement of G is chordal ([12]).
    Used in the introduction to explain when edge ideals have linear resolution, for example for C5.

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Pith. "Pith review of Gapfree graphs and powers of edge ideals with linear quotients." pith.science (2026). https://pith.science/paper/IIXOVXKC

@misc{pith2026241206467,
  author       = {Pith},
  title        = {Pith review of: Gapfree graphs and powers of edge ideals with linear quotients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IIXOVXKC}},
  note         = {Machine review of arXiv:2412.06467}
}
abstract

Let $I(G)$ be the edge ideal of a gapfree graph $G$. An open conjecture of Nevo and Peeva states that $I(G)^q$ has linear resolution for $q\gg 0$. We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if $I(G)^q$ has linear quotients for some integer $q\geq 1$, then $I(G)^{s}$ has linear quotients for all $s\geq q$. We give a partial solution to this conjecture, and identify conditions under which only finitely many powers need to be checked. It is known that if $G$ does not contain a cricket, a diamond, or a $C_4$, then $I(G)^q$ has linear resolution for $q \geq 2$. We construct a family of gapfree graphs $G$ containing cricket, diamond, $C_4$ together with $C_5$ as induced subgraphs of $G$ for which $I(G)^q$ has linear quotients for $q \ge 2$.

Figures

Figures reproduced from arXiv: 2412.06467 by the authors.

Figure 1
Figure 1. Cricket; Diamond; C4 For instance, the pentagon C5 is gapfree, but its complementary graph is not chordal. Consequently, I(C5) does not have linear resolution. However, C5 does not contain any induced subgraph that is isomorphic to a cricket (or a diamond, or C4), so I(C5) q has linear resolution for all q ≥ 2. On the other hand, it was shown (cf. [13, Corollary 10.1.7 and Theorems 10.1.9 and 10.2.5]) that if the co… view at source ↗
Figure 2
Figure 2. Graph G, Gx and G[x] We state some useful facts regarding the duplicated vertex x and its duplication y. The proofs are straightforward from the definitions and so are not included. First, recall that NG(x) = {z ∈ V (G) : xz ∈ E(G)} and NG[x] = NG(x) ∪ {x}. Lemma 3.3. Let G be a gapfree graph. Using the notation of Definition 3.1, • NGx(x) = NGx(y) = NG(x) = NG[x] \ {y}. • NG[x][x] = NG[x][y]. • G[x] is gapfree if a… view at source ↗
Figure 3
Figure 3. Pentagon together with one vertex e3e8 = e5e7, it follows that I(G) 2 has 8+1 2  − 2 = 36 − 2 = 34 generators. Let N (2) denote the following order on these generators: e 2 1 = (ab) 2 , e1e2 = (ab)(ax), e1e3 = (ab)(bx), e1e4 = (ab)(ap), e1e5 = (ab)(bq), e2e4 = (ax)(ap), e2e3 = (ax)(bx), e3e5 = (bx)(bq), e3e4 = (bx)(ap) e2e5 = (ax)(bq), e4e5 = (ap)(bq), e1e6 = (ab)(pz), e1e8 = (ab)(qz), e1e7 = (ab)(xz), e4e7 = (ap)(… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: (C5, K4) Example 5.3. Let I be the edge ideal of the graph (C5, Kn) for some n ≥ 1. Then I s has linear quotients for all s ≥ 2. The case n = 1 holds by Proposition 5.2. For n > 1 the result holds via induction using Proposition 2.3. Notice that the above argument cann…
Figure 5
Figure 5. Figure 5: Pentagon together with one vertex and multiple inner edges One can verify that I(G) 2 has linear quotients with respect to the following order: e 2 1 = (ab) 2 , e1e3 = (ab)(ax), e1e4 = (ab)(bx), e1e2 = (ab)(ap), e1e6 = (ab)(xp), e1e5 = (ab)(bq), e1e7 = (ab)(xq), e1e8 =…
Figure 6
Figure 6. Figure 6: A CDCC graph with minimum number of vertices 18 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

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  1. Powers of Edge Ideals with Linear Quotients

    math.AC 2024-12 accept novelty 8.0 of 10

    Explicit linear quotient orderings exist for all powers of anticycle edge ideals and for powers of any quadratic monomial ideal with linear quotients.

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