Green-Lazarsfeld Condition for Toric Edge Ideals of Bipartite Graphs

Jason McCullough; Zachary Greif

REVIEW 2 major objections 5 minor 1 cited by

Green-Lazarsfeld Condition for Toric Edge Ideals of Bipartite Graphs

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A bipartite graph's toric edge ideal is linearly presented exactly when its bipartite complement is essentially a tree of diameter at most 3, and the whole $\mathbf{N}_p$ ladder is classified.

desk verdict Completes the N_p classification for toric edge ideals of bipartite graphs; N2 proof is solid, but the N3 exceptional-characteristic claim rests on an unreproduced Macaulay2 computation. read the letter →

arxiv 1908.02744 v1 pith:GPABM7TU submitted 2019-08-07 math.AC math.AG

Zachary Greif , Jason McCullough This is my paper · ORCID

classification math.ACmath.AG MSC 13D0215A75

keywords Green-LazarsfeldconditiontoricedgeidealbipartitegraphlinearsyzygiesKoszulalgebracomplementpolyominogradedBettinumbers

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

Toric edge ideals are the defining ideals of edge rings, and the Green-Lazarsfeld condition $\mathbf{N}_p$ asks that the first $p$ steps of a minimal free resolution be linear. This paper completes the classification for bipartite graphs: each $\mathbf{N}_p$ becomes a purely graph-theoretic statement, from the known chord condition at $\mathbf{N}_1$ up to the single family $K_{2,n}$ at $\mathbf{N}_4$. The reason to care is that the proof turns a homological question into a finite list of forbidden induced subgraphs, and the classification has direct consequences for the regularity of these ideals and for polyomino ideals.

What carries the argument

The load-bearing objects are the bipartite complement $\overline{G}$ (same bipartition, edges reversed) and the notion of being essentially a tree (a tree after deleting isolated vertices). The engine is a local-to-global Betti obstruction: $\beta_{i,\alpha}(I_G)=\dim_k \tilde H_i(\Gamma(\alpha);k)$, where $\Gamma(\alpha)$ is the simplicial complex of monomials of multidegree $\alpha$; consequently every nonzero $\beta_{i,j}$ is witnessed by an induced subgraph on at most $2j$ vertices. This reduces each $\mathbf{N}_p$ failure to a finite forbidden-subgraph check. For $\mathbf{N}_2$, Proposition 5.1 — first syzygies in a Koszul ring are generated by linear syzygies and Koszul syzygies, the latter being the ones coming directly from pairs of quadratic relations — permits a five-case check of pairs of $4$-cycles, and Theorem 4.2 guarantees that whenever the bipartite complement is not essentially a tree of diameter at most $3$, one of the eight graphs $H^{(1)},\dots,H^{(8)}$ appears as an induced subgraph.

What would settle it

Compute the graded Betti table, over a field of characteristic not $3$, of the toric edge ideal of $K_{3,3}$ with one edge deleted; the paper's $\mathbf{N}_3$ argument predicts $\beta_{2,5}=1$, so a computation returning $\beta_{2,5}=0$ would refute the classification.

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Extended reading notes

Core claim

The central claim is Theorem 1.2. For a bipartite graph $G$ with minimum vertex degree at least $2$ over a field $k$, the toric edge ideal $I_G$ satisfies $\mathbf{N}_1$ exactly when every cycle of length at least $6$ has a chord; $\mathbf{N}_2$ exactly when the bipartite complement of $G$ is essentially a tree of diameter at most $3$; $\mathbf{N}_3$ exactly when $G$ is complete bipartite, except that $K_{m,n}$ with $\min\{m,n\}\ge 5$ fails $\mathbf{N}_3$ in characteristic $3$; and $\mathbf{N}_p$ for any $p\ge 4$ exactly when $G=K_{2,n}$. The proof route is: Theorem 3.2 turns a nonzero graded Betti number of $I_G$ into an induced subgraph on at most $2j$ vertices, Lemmas 3.3–3.6 exhibit the obstructions to $\mathbf{N}_2,\mathbf{N}_3,\mathbf{N}_4$, Theorem 4.2 converts the absence of eight forbidden induced subgraphs into the tree-complement statement, and Proposition 5.1 lets the $\mathbf{N}_2$ case be checked by examining the five configurations of two distinct $4$-cycles. The classification is then translated to convex polyomino ideals and used to rule out a family of linearly presented bipartite toric edge ideals with slow regularity growth.

Load-bearing premise

The $\mathbf{N}_2$ half of the theorem rests on the assumption that in a Koszul ring every first syzygy is generated by linear syzygies together with the syzygies coming directly from pairs of quadratic relations; if that decomposition misses a syzygy type, the proof's five-case check of pairs of $4$-cycles is incomplete.

Editorial extensions

If this is right

If $I_G$ satisfies $\mathbf{N}_2$, linear presentation can be read off by deleting isolated vertices from the bipartite complement and checking whether the rest is a tree of diameter at most $3$.
A failure of $\mathbf{N}_3$ for a non-complete bipartite graph shows up as a nonzero second syzygy in degree $5$, so only complete bipartite graphs can reach $\mathbf{N}_3$.
Once $\mathbf{N}_4$ holds, the graph is $K_{2,n}$ and the resolution is fully linear via the Eagon–Northcott complex, so $\mathbf{N}_4$ is the same as having a linear free resolution for these ideals.
For convex polyominoes, the translated criterion says $\mathbf{N}_2$ holds exactly when all missing cells lie in the first row or first column, $\mathbf{N}_3$ when the polyomino is an interval (with the same characteristic-$3$ caveat for large intervals), and $\mathbf{N}_4$ when the interval has one side of length $2$.
No family of bipartite graphs with linearly presented toric edge ideals can give a positive answer to the open question about linearly presented ideals with positive regularity-to-projective-dimension ratio: if the projective dimension tends to infinity, the ratio tends to $0$.

Reading between the lines

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

The same local-obstruction method suggests a finite decision procedure for $\mathbf{N}_p$ in any toric ring generated by quadrics: list the finitely many multidegrees of total degree $2j$ and check the reduced homology of their fibers; for bipartite graphs the list collapses to eight forbidden subgraphs, while for other normal toric rings it will be longer but still finite.
The $\mathbf{N}_2$ characterization doubles as a generator of examples: start with any tree of diameter at most $3$ on a fixed bipartition, take its bipartite complement, and the resulting graph has a linearly presented toric edge ideal, giving a large family on which regularity bounds can be tested.
The isolated characteristic-$3$ failure at $\mathbf{N}_3$ for large complete bipartite graphs hints that modular representation theory of symmetric groups controls higher linear syzygies of Segre varieties; the present classification shows such effects cannot survive past $\mathbf{N}_4$ for bipartite graphs, but they may appear in other toric ideals.
The paper's correction to the polyomino criterion illustrates a general caution: when a graph-theoretic theorem is transported through a dictionary such as convex polyominoes to bipartite graphs, configurations can be silently dropped, so the image of the dictionary should be rechecked against the full graph classification.

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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

Here's my take. The paper completes the N_p story for toric edge ideals of bipartite graphs, and the main theorem is a genuine classification: N2 iff the bipartite complement is essentially a tree of diameter at most 3, N3 iff complete bipartite except for the characteristic-3 K_{m,n} (min>=5) exception, and N4+ iff K_{2,n}. The N2 proof is the heart, and it holds up: the five configurations of two distinct 4-cycles are handled with explicit Koszul syzygy decompositions, and the graph-theoretic reduction in Theorem 4.2 is long but coherent. I don't see a missing case.

The paper also corrects a published result on linearly presented polyomino ideals [10], which is a nice bonus if it checks out.

The soft spot is real and it's the N3 theorem. Theorem 5.5 reduces to K_{5,5} and then asserts that 'a quick Macaulay2 calculation' shows p=3 is the only characteristic <20 where N3 fails. No script, no output, no alternative argument. Theorem 1.2(3) depends on exactly that computation. For p>20 the Lascoux-resolution argument is fine, and the characteristic-3 failure is backed by Hashimoto's result, but the exclusion of p=2,5,7,11,13,17,19 is the unreproduced part. That has to be fixed before I'd trust the statement as a theorem.

The N2 proof uses Proposition 5.1 from Mastroeni's Koszul paper. I found that application reasonable. The 'subscript inconsistency' the reader flagged looks like a labeling issue, not a mathematical gap: the two 4-cycles sharing two vertices are described with shared vertices on the Y side, so it's K_{4,2}, which is K_{2,4} up to swapping sides.

This is a paper I'd want to see in the literature. I'd send it to a referee with a request for the Macaulay2 script and the finite-field verification. It's not a desk reject; it's a solid result with one missing reproducibility check.

Referee Report

2 major / 5 minor

Summary. The paper characterizes the Green-Lazarsfeld conditions N_p for toric edge ideals of bipartite graphs with minimum degree at least 2. The main theorem gives a complete combinatorial description: N_1 is Ohsugi-Hibi chordality of cycles of length at least 6; N_2 holds exactly when the bipartite complement is essentially a tree of diameter at most 3; N_3 holds exactly for complete bipartite graphs except in characteristic 3 when both parts have size at least 5; and N_p for p at least 4 holds exactly for K_{2,n}. The proofs combine multigraded Betti-number obstructions and induced-subgraph reductions (Theorem 3.2), a graph-theoretic classification of the obstructions (Theorem 4.2), a Koszul-algebra reduction to linear and Koszul syzygies (Proposition 5.1 and the five-configuration analysis in Theorem 5.2), and known resolutions of Segre and determinantal ideals. The paper also translates the results to convex polyomino ideals, corrects a characterization in [10], and applies the N_2 result to a regularity question of Constantinescu, Kahle, and Varbaro.

Significance. If the results hold, they provide a complete and elegant combinatorial description of all Green-Lazarsfeld conditions for toric edge ideals of bipartite graphs, interpolating between Ohsugi-Hibi's quadratic-generation theorem and Ohsugi-Hibi's linear-resolution theorem. The N_2 proof supplies explicit syzygy decompositions for all five possible configurations of pairs of 4-cycles, and the global statement is checked against external benchmarks (Lascoux, Pragacz-Weyman, Hashimoto, Eagon-Northcott) rather than assumed. The polyomino application identifies an error in [10] and replaces it with a criterion that is consistent with the graph-theoretic classification. The regularity corollary for the Constantinescu-Kahle-Varbaro question is a clean consequence of the N_2 characterization. The main weakness is a load-bearing, unreproduced Macaulay2 computation in the proof of Theorem 5.5.

major comments (2)

[§5, proof of Theorem 5.5] The assertion that p=3 is the only characteristic less than 20 in which I_{K_{5,5}} fails to satisfy N_3 is load-bearing for Theorem 1.2(3), but the only evidence supplied is 'A quick Macaulay2 [22] calculation shows...' with no script, log, or reproducible output. The surrounding argument only proves failure in characteristic 3 and success in characteristic 0 and p>20; therefore, a second bad prime among 2,5,7,11,13,17,19 would falsify the theorem. Please provide the Macaulay2 code and output, or replace the computation with an invariant-theoretic or representation-theoretic argument. The sentence describing the p>20 reduction ('the number of boxes in the last partition...') is also too terse as written; the integral Lascoux complex should be shown explicitly to be a resolution over the relevant localized integer ring.
[§5, proof of Theorem 5.2 (Proposition 5.1)] The reduction of N_2 to checking five configurations of two distinct 4-cycles rests on Proposition 5.1, quoted from [21, Proposition 2.8], but the proposition is not stated in the paper and the term 'Koszul syzygy' is not defined. Because this proposition is load-bearing for the N_2 classification, the paper should state the proposition precisely, define the Koszul syzygies being considered, and justify that the five-configuration taxonomy together with the additional edges imposed by Theorem 4.2 exhausts all minimal Koszul syzygies in the relevant graphs. As written, a reader must take on faith both the validity of the quoted proposition in this setting and the exhaustiveness of the case analysis.

minor comments (5)

[Abstract vs. Theorem 1.2] The abstract states that N_2 holds if and only if the bipartite complement of G is a tree of diameter at most 3, while Theorem 1.2 and the main body state the condition as 'essentially a tree of diameter at most 3'; these statements should be aligned.
[Title and Section 6] There are several typographical errors: the title contains 'BIP ART ITE', Proposition 6.2 says 'every chord with length ≥ 6 has a chord' where 'cycle' is meant, and 'polynomo' appears in Proposition 6.2.
[Proof of Theorem 5.5] The phrase 'the only characteristic less than 20 which in which IG fails to satisfy N3' contains a duplicated 'which in'.
[Proposition 4.1] Proposition 4.1 as stated is not true for graphs with isolated vertices under the convention used in its proof, where a graph with no edges is counted as essentially connected; for example, a graph consisting of one edge plus an isolated vertex has every induced subgraph essentially connected and acyclic but is not a tree. The proposition is applied only to graphs without isolated vertices, so this does not affect the main theorem, but the statement should be restricted or the convention clarified.
[Section 6, after Remark 6.1] The sentence 'The ideal I_{H^{(1)}} ... is clearly not associated to any convex polyomino, which cannot have exactly 2 minimal generators' is ambiguous: 'which' appears to refer to a convex polyomino ideal rather than to a polyomino. Please rephrase for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the paper derives new equivalences from external cited theorems and independent Betti-number and graph-theoretic arguments.

full rationale

The derivation chain is self-contained in the relevant sense. Theorem 1.2(1) is a restatement of Ohsugi–Hibi (Theorem 1.1), an externally cited result; Theorem 5.2 proves the N2 classification using the external Koszul-syzygy decomposition [21, Proposition 2.8], a purely graph-theoretic classification Theorem 4.2, and explicit induced-subgraph Betti computations in Lemmas 3.3–3.4. No parameter is fitted, and no 'prediction' is set equal to an input by construction. Theorem 5.5 uses the external Lascoux/Pragacz–Weyman resolutions and [2, Main Theorem (2)]; the only unexpanded step is the sentence 'A quick Macaulay2 [22] calculation shows that p = 3 is the only characteristic less than 20 which in which IG fails to satisfy N3.' That is a finite computational check, and although it is load-bearing and not reproducible from the text, it is a computation about the same object being classified, not an assumption of the target equivalence. It is therefore a correctness/reproducibility risk, not circularity. Theorem 5.6 is likewise a short reduction to K3,3 and the Eagon–Northcott resolution. There are no author self-citations carrying the argument, no uniqueness theorem imported from the same authors, and no ansatz smuggled in by citation. The correction to [10] in Section 6 is a translation of the independently proved N2 theorem to polyomino ideals, not a renaming of a known result as a new one.

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

No free parameters or invented entities appear; the contribution is a mathematical theorem. The paper's inputs are standard results from the literature plus one unshipped computer check.

assumptions (5)

standard math Ohsugi-Hibi Theorem 1.1: N1 iff every cycle of length at least 6 has a chord, equivalent to a quadratic Grobner basis and Koszul property.
Used as the starting point for Theorem 1.2(1) and as the N1 hypothesis throughout Section 5.
standard math Mastroeni Proposition 5.1: in a Koszul algebra, the first syzygies of I are minimally generated by linear syzygies and Koszul syzygies.
Central reduction in the proof of Theorem 5.2 that limits the check to pairs of 4-cycles.
standard math Avramov-Conca-Iyengar Main Theorem (2): for linearly presented Koszul algebras, beta_{2,j} vanishes for all j at least 6.
Used in Theorem 5.5 to reduce failure of N3 to a nonzero beta_{2,5}.
standard math Lascoux and Pragacz-Weyman resolutions of determinantal ideals, with characteristic independence results of Hashimoto-Kurano and Akin-Buchsbaum-Weyman.
Used to assert that complete bipartite graphs satisfy N3 outside characteristic 3 and to describe the K_{2,n} resolution.
domain assumption Macaulay2 computation showing that p=3 is the only characteristic below 20 for which K5,5 fails N3.
No script, input, or output is shipped; the finite computational check is asserted in the text.

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Pith. "Pith review of Green-Lazarsfeld Condition for Toric Edge Ideals of Bipartite Graphs." pith.science (2026). https://pith.science/paper/GPABM7TU

@misc{pith2026190802744,
  author       = {Pith},
  title        = {Pith review of: Green-Lazarsfeld Condition for Toric Edge Ideals of Bipartite Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GPABM7TU}},
  note         = {Machine review of arXiv:1908.02744}
}

abstract

Previously, Ohsugi and Hibi gave a combinatorial description of bipartite graphs $G$ whose toric edge ideal $I_G$ is generated by quadrics, showing that every cycle of $G$ of length at least $6$ must have a chord. This corresponds to the Green-Lazarsfeld condition $\mathbf{N}_1$. In this paper, we investigate the higher syzygies of $I_G$ and give combinatorial descriptions of the Green-Lazarsfeld conditions $\mathbf{N}_p$ of toric edge ideals of bipartite graphs for all $p \ge 1$. In particular, we show that $I_G$ is linearly presented (i.e. satisfies condition $\mathbf{N}_2$) if and only if the bipartite complement of $G$ is a tree of diameter at most $3$. We also investigate the regularity of linearly presented toric edge ideals and give criteria for polyomino ideals to satisfy the Green-Lazarsfeld conditions.

Figures

Figures reproduced from arXiv: 1908.02744 by the authors.

**Figure 1.** Induced subgraphs that are obstructions to satisfying condition N2 Lemma 3.3. IH(1) and IH(8) do not satisfy condition N2 and in particular β1,4(IH(1) ) and β1,4(IH(8) ) are nonzero. Proof. The ideals IH(1) and IH(8) are complete intersections generated by two quadrics. In particular β1,4(IH(1) ) = 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** A visualization of Γ(α) in which each shaded tetrahedron represents a facet. The geometric realization of the abstract simplicial complex is Γ(α) contracts to a circle. Thus β1,α(IH(i) ) = dimk He1(Γ(α); k) = 1. By Theorem 3.2, β1,4(IH(i) ) 6= 0 for i = 2, . . . , 7 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 8.** Three examples of bipartite graphs with non-linearly presented toric edge ideals and their Betti tables Consider the complete bipartite graph K5,5. Then β2,5(IK5,5 ) = 0 if and only if the characteristic of k is not 3. In particular, in characteristic other than 3, IK5,5 has partial graded Betti table 0 1 2 2: 100 800 3075 · · · , while in characteristic 3, IK5,5 has partial graded Betti table 0 1 2 2: 100 800 3075 … view at source ↗

Figures from the paper (2 more)

**Figure 10.** Figure 10: A linearly related polyomino [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

**Figure 11.** Figure 11: A general, linearly related, convex polyomino. 7. Application to a Question of Constantinescu, Kahle, and Varbaro Our original motivation for studying linearly presented toric edge ideals comes from the following question of Constantinescu, Kahle, and Varbaro [6]: Que…

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Edge rings of bipartite graphs with linear resolutions
math.AC 2019-08 accept novelty 7.0 of 10

A connected bipartite graph whose edge ring has a q-linear resolution, q≥3, has exactly one minimal even cycle, so its edge ring is a hypersurface.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [10]

V. Ene, J. Herzog, T. Hibi, Linearly related polyominoe s. J. Algebr. Comb. (2015) 41:949–968

work page 2015

[2] [22]

Grayson, and M

D. Grayson, and M. Stillman, Macaulay2, a software syst em for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/

work page

[3] [1]

K. Akin, D. Buchsbaum, J. W eyman, Resolutions of determi nantal ideals: the submaximal minors. Adv. in Math. 39 (1981), no. 1, 1–30

work page 1981

[4] [2]

Avramov, A

L. Avramov, A. Conca and S. Iyengar, Free resolutions ove r commutative Koszul algebras. Math. Res. Lett. 17 (2010), no. 2, 197–210

work page 2010

[5] [3]

Biermann, A

J. Biermann, A. O’Keefe, A. Van Tuyl, Bounds on the regula rity of toric ideals of graphs. Adv. in Appl. Math. 85 (2017), 84–102

work page 2017

[6] [4]

Buchsbaum, D

D.A. Buchsbaum, D. Eisenbud, Algebra structures for ﬁni te free resolutions, and some structure theorems for ideals of codimension 3, Am. J. Math. 99 (3) (1977) 447–485

work page 1977

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Conca, Koszul algebras and their syzygies

A. Conca, Koszul algebras and their syzygies. Combinato rial algebraic geometry, 1–31, Lecture Notes in Math., 2108 , Fond. CIME/CIME Found. Subser., Springer, Cham, 2014

work page 2014

[8] [6]

Constantinescu, T

A. Constantinescu, T. Kahle, M. Varbaro, Linear syzygie s, hyperbolic Coxeter groups and regularity. Compos. Math. 155 (2019), no. 6, 1076–1097

work page 2019