The $h$-vectors of toric ideals of odd cycle compositions revisited

Adam Van Tuyl; Kieran Bhaskara; Sasha Zotine

REVIEW 2 major objections 4 minor 14 references

The $h$-vectors of toric ideals of odd cycle compositions revisited

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The toric ideal of s odd cycles sharing a common vertex is geometrically vertex decomposable.

desk verdict A short, mostly correct paper proving a stronger property than the known h-polynomial result, but the y-compatibility argument needs a fix before it is fully rigorous. read the letter →

arxiv 2504.13087 v1 pith:YCMWVRPV submitted 2025-04-17 math.AC math.CO

Kieran Bhaskara , Adam Van Tuyl , Sasha Zotine This is my paper · ORCID

classification math.ACmath.CO MSC 13D0213D4013P1013F6514M2505E40

keywords geometricallyvertexdecomposabletoricidealofagraphh-polynomialoddcyclecompositionuniversalGrobnerbasisCastelnuovo-Mumfordregularitysquarefreemonomialcompleteintersectionedgering

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 sets out to prove that the toric ideal of a graph built from s odd cycles that share a single common vertex is geometrically vertex decomposable. That property is stronger than knowing the h-polynomial, because it packages the ideal as a recursive sequence of two smaller ideals, ending in a complete intersection. If the proof is correct, the known h-polynomial formula for these edge rings follows as a corollary, along with a new formula for the Castelnuovo-Mumford regularity. The proof avoids the initial-ideal and simplicial-complex route of earlier work.

What carries the argument

The central object is the geometrically vertex decomposable ideal, an ideal that can be split recursively as iny(I) = C ∩ (N + ⟨y⟩) with both contracted pieces GVD, and whose bottom cases are variable ideals or the unit ideal. The workhorse is the pair of ideals Ci and Ni generated from the universal Gröbner basis of IG; the basis itself is described by primitive closed even walks that pass through exactly two odd cycles. Each step peels one odd-indexed edge off the last cycle, Ni stays equal to the toric ideal of the graph with s − 1 cycles, and the terminal ideal C{ts} is generated by even-indexed edges of the earlier cycles, forming a squarefree monomial complete intersection.

What would settle it

Compute the primitive closed even walks of G(1,1,1), the graph of three triangles sharing a vertex. If any such walk visits all three cycles, the corresponding binomial is missing from the set in Lemma 3.2, and the C/N decomposition would not be a Gröbner basis; that would block the induction.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for G = G(t1, . . . , ts), the toric ideal IG is geometrically vertex decomposable. Using the universal Gröbner basis of Lemma 3.2, the paper fixes a lexicographic order that puts odd-indexed edges of the last cycle on top, then builds ideals Ci and Ni by taking initial forms. It verifies the defining intersection condition iny(I) = Ci ∩ (Ni + ⟨y⟩) via the criterion of Lemma 2.2, observes that every Ni equals IG(t1,...,t{s-1}), and shows the final ideal C{ts} is a squarefree monomial complete intersection. Induction on the number of cycles then gives the decomposition for all s. The h-polynomial formula and the regularity formula are consequences of this structural statement rather than separate computations.

Load-bearing premise

The construction stands or falls on the assertion in Lemma 3.2 that every primitive closed even walk of G(t1, . . . , ts) runs through exactly two of the odd cycles; the proof is a sketch that delegates the classification to a prior paper.

Editorial extensions

If this is right

The h-polynomial of K[G(t1, . . . , ts)] equals $\prod_{i=1}^s (1+z+\cdots+z^{t_i}) - z \prod_{i=1}^s (1+z+\cdots+z^{t_i-1})$.
The Castelnuovo-Mumford regularity is $t_1+\cdots+t_s$ when $s \ge 2$, and $0$ when $s=1$.
Each ring K[G(t1, . . . , ts)] is Cohen-Macaulay, since GVD ideals are Cohen-Macaulay.
The GVD construction gives a uniform inductive framework: the N ideals are exactly the toric ideals of the smaller composition, so all numerical invariants propagate by the same recursion.

Reading between the lines

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

The same C/N peeling may yield explicit free resolutions or Betti numbers for these edge rings, since GVD decompositions often carry resolution information beyond Hilbert-series data.
If the primitive-walk classification in Lemma 3.2 generalizes, the method could extend to graphs whose odd cycles share a path or a block rather than a single vertex.
The terminal complete intersection C{ts}, generated by even-indexed edges of all but the last cycle, isolates the highest-degree part of the h-polynomial; studying it separately might explain the regularity formula without invoking Cohen-Macaulayness.

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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

Quick take: the main result is real and the proof is almost right, but the y-compatibility step in the GVD induction is written incorrectly. The paper proves that toric ideals of graphs consisting of odd cycles sharing a vertex are geometrically vertex decomposable, which is genuinely new and stronger than the previously known h-polynomial formula. The regularity formula is also new. The recovery of the h-polynomial as a corollary is a nice illustration of the GVD machinery.

What I like: the C/N decomposition is clean, the explicit example for G(2,1,2) helps, and the induction on both number of cycles and number of odd-indexed edges is well organized. The universal Grobner basis claim in Lemma 3.2 is given as a sketch, but I think the two-cycle-only classification is correct: a primitive even closed walk cannot pass through three or more cycles because the prefix ending after two cycles is already a closed even sub-walk. So that concern is minor.

The soft spot is the monomial order. The proof fixes a lex order with the odd edges of the last cycle in decreasing order, and then claims the order is always y-compatible for each successive y. That is not true in the ambient ring: after the first split, y = e_{s,2t_s-1} is not the largest variable, so for f = e_{s,2t_s+1} + e_{s,2t_s-1}, the leading term and the initial y-form disagree. The fix is to pass to the subring with the already eliminated variables contracted away, where the current y is indeed largest. The ideals C_i appear to live in that subring, so the argument can be repaired, but the paper does not say this and the Grobner basis statements need to be rechecked in the subring setting. As written, Lemma 2.2 is invoked without the required hypothesis.

There are also small omissions, like not verifying the hypotheses of Theorem 2.5 in the h-polynomial proof, but those are easy and likely fine.

Bottom line: this is a useful, modest contribution for people working on geometric vertex decompositions and toric ideals of graphs. It deserves a serious referee; I would accept it with a request to fix the y-compatibility gap and expand the sketch in Lemma 3.2 a bit.

Referee Report

2 major / 4 minor

Summary. The paper studies the toric ideal I_G of the graph G = G(t_1, ..., t_s) consisting of s odd cycles sharing a common vertex. It proves Theorem 1.1, asserting that I_G is geometrically vertex decomposable (GVD), and then derives Corollary 1.2, the h-polynomial formula previously obtained by Bhaskara, Higashitani, and Shibu Deepthi, as well as Corollary 1.3, which computes the Castelnuovo-Mumford regularity. The proof proceeds by induction on the number of cycles and, within the last cycle, by induction on the odd-indexed edges. For each such edge the authors define ideals C_i and N_i via equations (3.2)-(3.3), invoke Lemma 2.2 to obtain the GVD intersection condition, and observe that the terminal ideal C_{t_s} is a squarefree monomial complete intersection. The paper is a short note whose main contribution is the stronger GVD property and the resulting conceptual proof of the known h-polynomial formula.

Significance. If Theorem 1.1 is correct, it gives a structural explanation for the h-polynomial of these toric edge rings and provides a new route to the regularity formula, avoiding the initial-ideal computations of [2]. The inductive decomposition is elegant: the N-ideals are the toric ideals of the smaller graph G(t_1, ..., t_{s-1}), and each C-chain terminates in a squarefree monomial complete intersection. The paper also carefully identifies the universal Grobner basis of I_G (Lemma 3.2), which is a useful explicit description. These are genuine strengths. However, the proof currently has a load-bearing gap concerning the y-compatibility of the fixed monomial order, and the universal-Grobner-basis lemma is only sketched. Since the central claim appears defensible and the gap is likely repairable, the appropriate outcome is major revision rather than rejection.

major comments (2)

[Section 3, proof of Theorem 1.1 (after Eq. (3.1))] The fixed lexicographic order with e_{s,2t_s+1} > e_{s,2t_s-1} > ... > e_{s,1} > remaining variables is claimed to be 'always y-compatible' for the successive choices y = e_{s,2(t_s-i)+1}. This is false for i >= 1. For example, take y = e_{s,2t_s-1} and f = e_{s,2t_s+1} + e_{s,2t_s-1}; then in_y(f) = e_{s,2t_s-1}, so in_<(in_y(f)) = e_{s,2t_s-1}, but in_<(f) = e_{s,2t_s+1}. This contradicts the definition of y-compatible order in Section 2. Consequently Lemma 2.2 cannot be invoked to justify the equalities in_y(C_{i-1}) = C_i cap (N_i + <y>) for i >= 1, and the GVD induction does not go through as written. The argument can likely be repaired by choosing a fresh y-compatible order at each decomposition step (with the current y as the largest variable) and re-verifying that the displayed generating sets remain Grobner bases for that order, but this repair is not present in the manuscript.
[Section 3, Lemma 3.2] Lemma 3.2 asserts that the set in equation (3.1) is a universal Grobner basis of I_G, but the proof is only a sketch. The key step, the classification of primitive closed even walks of G(t_1, ..., t_s), is delegated to [2] with the statement that every such walk runs through exactly two of the odd cycles. Since this set is used as the Grobner basis in Lemma 2.2 at every decomposition step, Theorem 1.1 depends on this classification. The reader's worry about primitive walks through three cycles is not decisive, because each odd cycle contributes ±2 at the common center and circulations decompose into two-cycle alternating walks; nevertheless, the manuscript should either give a complete proof of Lemma 3.2 or state and cite the precise theorem from [2] that supplies the classification, rather than a sketch.

minor comments (4)

[Abstract] The word 'h-polyhomial' in the abstract is a typo and should read 'h-polynomial'.
[Section 2] In the paragraph after Definition 2.1, 'resect to' should be 'respect to'.
[Example 3.3 and proof of Theorem 1.1] The notation N is reused for N_{e3,5,I}, N_{e3,3,C0}, and N_{e3,1,C1}; since all these ideals are equal, it would be clearer to define N := N_0 once and use it throughout, as is done later in the proof of Corollary 1.2.
[Proof of Corollary 1.2] When Theorem 2.5 is applied successively to the ideals C_i, the hypotheses sqrt(C_{y,C_i}) != sqrt(N) and C_{y,C_i} != <1> are not verified explicitly. They are plausible because C_{i+1} contains monomials in the last-cycle variables whereas N does not, but a one-sentence check would make the argument complete.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the GVD proof is an independent derivation; the h-polynomial is re-derived, not assumed, and the cited walk classification in [2] is external support.

full rationale

The central claim (Theorem 1.1) is proved by an induction that constructs ideals C_i and N_i from the generating set in Lemma 3.2 and verifies the GVD intersection condition via Lemma 2.2. The h-polynomial of Corollary 1.2 is not used as an input; it is re-derived in the proof of Corollary 1.2 from Theorem 2.5, the induction hypothesis, and the complete-intersection computation for C_{t_s}. Corollary 1.3 follows from Cohen-Macaulayness of GVD ideals and the degree of the h-polynomial, again not from a fitted or pre-supplied value. The only load-bearing external input is Lemma 3.2's universal Grobner basis, whose proof is a sketch and delegates the classification of primitive closed even walks to [2]. This is a self-citation, since Bhaskara is an author of both [2] and the present paper, and the lemma is load-bearing. However, [2]'s classification is a distinct, parameter-free combinatorial statement about the same graph family; it does not include the GVD or h-polynomial conclusions. Under the stated rules, that is independent support rather than a circular step. There are no fitted parameters, no predictions that equal their inputs by construction, and no renaming of a known result as a new one. The proof sketch in Lemma 3.2 and the fixed lex order's y-compatibility for later splits are potential correctness concerns, but they are proof gaps, not circularity. The derivation chain therefore has no significant circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claim rests on standard Hilbert-series and GVD machinery, plus a classification of primitive closed even walks delegated to [2]. There are no fitted constants, no data, and no invented entities.

assumptions (9)

standard math Hilbert-Serre theorem: Hilbert series of a standard graded ring is a rational function with denominator (1-z)^n.
Used in Section 2 to define the h-polynomial as the numerator of the reduced Hilbert series.
standard math Knutson-Miller-Yong Lemma 2.2: if a Grobner basis has initial y-forms with d=0 or 1, then in_y(I)=C cap (N + <y>) and the q_i form Grobner bases.
Applied repeatedly in the proof of Theorem 1.1 to obtain the C_i/N_i decompositions.
standard math Squarefree monomial complete intersections are geometrically vertex decomposable (Lemma 2.3, cited from [5]).
Used to conclude that C_{t_s} is GVD.
standard math Complete intersection h-polynomial formula (Lemma 2.4): h(R/I) is the product over generators of (1 + z + ... + z^{d_i-1}).
Used to compute h(R/C_{t_s}) in Corollary 1.2.
standard math GVD h-polynomial recurrence of Nguyen-Rajchgot-Van Tuyl (Theorem 2.5): h(R/I)=h(R/N)+z h(R/C).
Basis for the recursive computation in Corollary 1.2.
standard math Primitive closed even walks of a graph give a universal Grobner basis of the toric ideal (Villarreal, Prop 10.1.10).
Used in Lemma 3.2 to reduce the Grobner basis to the displayed binomials.
domain assumption Classification of primitive closed even walks of G(t1,...,ts): every such walk traverses exactly two of the odd cycles (claimed from [2]).
Explicitly only sketched in Lemma 3.2 and delegated to [2], a paper with overlapping authorship; this is the load-bearing premise behind the generating set (3.1).
standard math GVD ideals are Cohen-Macaulay (Klein-Rajchgot, Corollary 4.5).
Used in Corollary 1.3 to convert the h-polynomial into the regularity via reg(R/I)=deg h(R/I).
ad hoc to paper The lexicographic order with e_{s,2t_s+1} > e_{s,2t_s-1} > ... > e_{s,1} > remaining variables is y-compatible for each selected y.
Asserted 'by construction' in the proof of Theorem 1.1; needed for every application of Lemma 2.2.

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Pith. "Pith review of The $h$-vectors of toric ideals of odd cycle compositions revisited." pith.science (2026). https://pith.science/paper/YCMWVRPV

@misc{pith2026250413087,
  author       = {Pith},
  title        = {Pith review of: The $h$-vectors of toric ideals of odd cycle compositions revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YCMWVRPV}},
  note         = {Machine review of arXiv:2504.13087}
}

abstract

Let $G$ be a graph consisting of $s$ odd cycles that all share a common vertex. Bhaskara, Higashitani, and Shibu Deepthi recently computed the $h$-polynomial for the quotient ring $R/I_G$, where $I_G$ is the toric ideal of $G$, in terms of the number and sizes of odd cycles in the graph. The purpose of this note is to prove the stronger result that these toric ideals are geometrically vertex decomposable, which allows us to deduce the result of Bhaskara, Higashitani, and Shibu Deepthi about the $h$-polyhomial as a corollary.

Figures

Figures reproduced from arXiv: 2504.13087 by the authors.

**Figure 1.** The graph G(2, 1, 2) consisting of two 2 · 2 + 1 = 5-cycles and one 2 · 1 + 1 = 3-cycle meeting at a common vertex main result of [2] is to compute the h-polynomial of K[G(t1, . . . , ts)], that is, the numerator of the reduced Hilbert series of K[G(t1, . . . , ts)] (see the next section or [2] for any undefined Date: April 18, 2025. 2020 Mathematics Subject Classification. 13D02, 13D40, 13P10, 13F65, 14M25, 05E40. … view at source ↗

**Figure 2.** The graph G = G(t1, t2, . . . , ts) and the notation for its edges. Remark 3.1. Note that our notation for the graph G(t1, . . . , ts) is different than that found in [2]. In that paper the authors write G(r1, . . . , rt) where ri is the number of odd-cycles of length 2i + 1. We also want to highlight that we make no assumptions on the order of t1, . . . , ts. However, it should be clear that for any permutation σ ∈… view at source ↗

**Figure 3.** Diagram summarizing the argument for why IG(t1,...,ts) is GVD. Proof of Theorem 1.1. We perform induction on the number of cycles s. The base case is s = 1, in which case the toric ideal is trivial and hence GVD. Assume s ≥ 2 and let G = G(t1, t2, . . . , ts). Lemma 3.2 gives us that the toric ideal of G is (3.1) I := IG(t1,...,ts) = *Y tp i=0 ep,2i+1Y tq j=1 eq,2j − Y tq k=0 eq,2k+1Y tp ℓ=1 ep,2ℓ [PITH_F… view at source ↗

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Works this paper leans on

14 extracted references · 14 canonical work pages

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [2]

The h-vectors of edge rings of odd cycle compositions

Kieran Bhaskara, Akihiro Higashitani, and Nayana Shibu Deepthi. The h-vectors of edge rings of odd cycle compositions. J. Algebra Appl. , 2023. paper no. 2550362, 2025

work page 2023

[2] [1]

Root pol ytopes, tropical types, and toric edge ideals

Ayah Almousa, Anton Dochtermann, and Ben Smith. Root pol ytopes, tropical types, and toric edge ideals. Algebr. Comb. , 8(1):59–99, 2025

work page 2025

[3] [3]

Bounds on the regularity of toric ideals of graphs

Jennifer Biermann, Augustine O’Keefe, and Adam Van Tuyl . Bounds on the regularity of toric ideals of graphs. Adv. in Appl. Math. , 85:84–102, 2017

work page 2017

[4] [4]

Cohen–Macaulay rings , volume 39 of Cambridge Studies in Advanced Mathematics

Winfried Bruns and Jürgen Herzog. Cohen–Macaulay rings , volume 39 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1993

work page 1993

[5] [5]

Geometric vertex decomposition and liaison for toric ideals of graphs

Mike Cummings, Sergio Da Silva, Jenna Rajchgot, and Adam Van Tuyl. Geometric vertex decomposition and liaison for toric ideals of graphs. Algebr. Comb. , 6(4):965–997, 2023

work page 2023

[6] [6]

Geometric Vertex Deco mposition, Gröbner bases, and Frobenius Splittings for Regular Nilpotent Hessenberg Varieties

Sergio Da Silva and Megumi Harada. Geometric Vertex Deco mposition, Gröbner bases, and Frobenius Splittings for Regular Nilpotent Hessenberg Varieties. Transformation Groups, pages 1–36, 2023

work page 2023

[7] [7]

Algebraic properties of toric rings of graphs

Huy Tài Hà, Selvi Kara Beyarslan, and Augustine O’Keefe. Algebraic properties of toric rings of graphs. Comm. Algebra, 47(1):1–16, 2019

work page 2019

[8] [8]

Takayuki Hibi, Akihiro Higashitani, Kyouko Kimura, and Augustine B. O’Keefe. Depth of edge rings arising from ﬁnite graphs. Proc. Amer. Math. Soc. , 139(11):3807–3813, 2011

work page 2011