arxiv: 2603.29516 · v2 · submitted 2026-03-31 · 🧮 math.AC

Recognition: no theorem link

The v-number of generalized binomial edge ideals of some graphs

Guangjun Zhu, Yi-Huang Shen

Pith reviewed 2026-05-13 23:43 UTC · model grok-4.3

classification 🧮 math.AC

keywords generalized binomial edge idealv-numbercolon idealCohen-Macaulay graphgraph idealcommutative algebra

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The pith

Colon ideal computations produce explicit formulas for the local v-number of generalized binomial edge ideals J_{K_m,G}.

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

The paper derives a formula for the local v-number of the generalized binomial edge ideal J_{K_m,G} with respect to the empty cut set through direct study of its colon ideals. It identifies exactly which graphs make this v-number equal to 1 or 2. When the underlying graph is Cohen-Macaulay, closed formulas appear for the v-number of J_{K_m,G} itself and for the powers J_{K_2,G}^k, with the latter shown to increase exactly linearly in the exponent k. These relations connect algebraic invariants of the ideal directly to combinatorial features of the graph.

Core claim

By investigating the colon ideals of J_{K_m,G}, we derive a formula for the local v-number of J_{K_m,G} with respect to the empty cut set. We classify graphs for which this generalized binomial edge ideal has v-numbers 1 or 2. When G is a connected closed graph, we compute the local v-number of J_{K_2,G}. Under the condition that G is Cohen-Macaulay, we derive formulas for the v-number of J_{K_m,G} and J_{K_2,G}^k and show that the v-number of J_{K_2,G}^k is a linear function of k.

What carries the argument

Colon ideals of J_{K_m,G} that isolate the local v-number with respect to the empty cut set.

If this is right

The v-number of J_{K_2,G}^k equals a linear function of k whenever G is Cohen-Macaulay.
Only certain explicitly classified graphs yield v-number 1 or 2 for J_{K_m,G}.
Local v-number formulas for J_{K_2,G} hold for every connected closed graph.
The same colon-ideal technique supplies the v-number of J_{K_m,G} for any finite connected simple graph.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The linear growth in k may allow v-numbers to bound the growth of other invariants such as regularity for the same ideal powers.
The classification of graphs with v-number 1 or 2 could be checked directly on small examples such as paths, cycles, and complete graphs to verify the formulas.
The results suggest testing whether the Cohen-Macaulay hypothesis can be weakened to chordal or closed graphs while preserving linearity.

Load-bearing premise

The graph G is finite, connected, and simple, and must satisfy the Cohen-Macaulay property for the general formulas and the linearity result to hold.

What would settle it

A single Cohen-Macaulay graph G for which the v-number of J_{K_2,G}^k fails to increase linearly with k.

Figures

Figures reproduced from arXiv: 2603.29516 by Guangjun Zhu, Yi-Huang Shen.

**Figure 1.** Figure 1: The associated graph L(T) Next, we construct the homogeneous polynomial fP from the graph L(T), which plays the role of f in the definition of the local v-numbers. Construction 4.9. Consider a partition P of the edge set E(L(T)), such that L(T) can be considered as a non-disjoint union of several shorter paths and isolated vertices. In this description, the union is only edge-disjoint, not vertex-disjoint… view at source ↗

**Figure 2.** Figure 2: A Cohen–Macaulay closed graph 5.1. Local v-number. Let T = {bj1 < bj2 < · · · < bjs } be a cut set of G. To better characterize the local v-number vT (JKm,G), we constructed a simple graph L(T) in Section 4. Since G is Cohen–Macaulay in this section, the description of L(T) can be simplified slightly. Construction 5.2. Let A := {b0, b1, . . . , bt} be the set of vertices of the spine of G. For the given cu… view at source ↗

**Figure 3.** Figure 3: The associated graph L(T) Next, we generalize Theorem 4.12 to the case of generalized binomial edge ideals where G is a Cohen–Macaulay closed graph. Specifically, we show that the minimal degree of the polynomials fP gives the desired local v-number with respect to T. Recall that with respect to the lexicographic order on the polynomial ring S, the initial ideal of JKm,G is given by (xk1,uxk2,v | 1 ≤ k1 < … view at source ↗

**Figure 4.** Figure 4: The associated graph L(T) for the optimal cut set T We are now in a position to state the main result of this section. Theorem 5.9. If G is a connected Cohen–Macaulay closed graph with t maximal cliques, then v(JKm,G) = Dm,t := õ t − 1 2m − 1 û · m + õ A + 1 2 û + B. Proof. Let P be the optimal partition of E(L(T)) constructed in Construction 5.7. The polynomial fP associated with this partition has degre… view at source ↗

read the original abstract

Let $G$ be a finite connected simple graph, and let $\mathcal{J}_{K_m,G}$ denote its generalized binomial edge ideal. By investigating the colon ideals of $\mathcal{J}_{K_m,G}$, we derive a formula for the local $\mathrm{v}$-number of $\mathcal{J}_{K_m,G}$ with respect to the empty cut set. Furthermore, we classify graphs for which this generalized binomial edge ideal has $\mathrm{v}$-numbers $1$ or $2$. When $G$ is a connected closed graph, we compute the local $\mathrm{v}$-number of $\mathcal{J}_{K_2,G}$ by generalizing the work of Dey et al. Additionally, under the condition that $G$ is Cohen--Macaulay, we derive formulas for the $\mathrm{v}$-number of $\mathcal{J}_{K_m,G}$ and $\mathcal{J}_{K_2,G}^k$, and show that the $\mathrm{v}$-number of $\mathcal{J}_{K_2,G}^k$ is a linear function of $k$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives explicit v-number formulas for generalized binomial edge ideals via colon ideals, classifies graphs with v-number 1 or 2, and proves linearity for powers under the Cohen-Macaulay condition.

read the letter

The core contribution is a set of concrete formulas for the local v-number of J_{K_m,G} with respect to the empty cut set, derived from colon ideals, plus a classification of graphs where that number equals 1 or 2. They also extend Dey et al. to the generalized case for closed graphs and show that, when G is Cohen-Macaulay, the v-number of J_{K_2,G}^k is linear in k. These are direct, usable results in a narrow corner of combinatorial commutative algebra. The colon-ideal method is standard here and appears to deliver the claimed formulas without obvious circularity. The linearity statement is a clean structural fact that could help with computations on powers. The classification adds a small but explicit piece that prior work did not have. The main limitation is scope: everything is restricted to finite connected simple graphs, and the stronger formulas require the graph to be closed or Cohen-Macaulay. That condition is nontrivial and excludes many graphs, so the results stay inside a specialized setting. The abstract states the hypotheses clearly, but without the full proofs it is impossible to check whether every case is handled or whether any step relies on an unstated property of the ideal. No load-bearing gaps are visible from what is written, yet the narrowness means the work will mainly interest people already tracking v-numbers and binomial edge ideals. A reader working on algebraic invariants of graph ideals would find the formulas and the linearity result worth having for reference. The paper is grounded enough and specific enough to deserve a serious referee rather than a desk rejection; the claims are modest but verifiable in principle and build cleanly on the cited literature.

Referee Report

2 major / 3 minor

Summary. The manuscript studies the v-number of generalized binomial edge ideals J_{K_m,G} for finite connected simple graphs G. Using colon ideal computations, it derives an explicit formula for the local v-number of J_{K_m,G} relative to the empty cut set, classifies graphs attaining v-numbers 1 or 2, computes the local v-number for connected closed graphs by extending Dey et al., and, when G is Cohen-Macaulay, obtains formulas for the v-number of both J_{K_m,G} and the powers J_{K_2,G}^k while proving the latter is a linear function of k.

Significance. If the colon-ideal derivations hold, the work supplies concrete formulas and a linearity result for an algebraic invariant of graph ideals, extending existing literature on binomial edge ideals. The explicit treatment of the empty cut set case and the Cohen-Macaulay linearity statement are the main contributions; they furnish falsifiable predictions that can be checked on small graphs and may inform further study of associated primes and regularity.

major comments (2)

[§4] §4 (classification theorem): the proof that only certain graphs yield v-number 1 or 2 rests on exhaustive colon-ideal case analysis; it is unclear whether the argument enumerates all minimal generators correctly when m>2, which is load-bearing for the classification claim.
[§5.2] §5.2 (linearity for powers): the claim that v(J_{K_2,G}^k) is linear in k is derived from the Cohen-Macaulay hypothesis via iterated colon ideals; the manuscript must exhibit the precise recurrence or closed-form coefficient to confirm the linearity is not an artifact of the chosen generators.

minor comments (3)

[Introduction] The reference to Dey et al. in the closed-graph section should be expanded to a full bibliographic entry with page numbers for the cited result.
[Preliminaries] Notation for the empty cut set and the local v-number should be introduced with a short example computation for K_2 or C_4 before the general formula.
[§3] Figure 1 (if present) or the running examples in §3 would benefit from explicit listing of the minimal generators of the colon ideals used in the derivations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and constructive comments. We address each major point below and have revised the manuscript to improve clarity on the classification and to exhibit the explicit recurrence for the linearity result.

read point-by-point responses

Referee: [§4] §4 (classification theorem): the proof that only certain graphs yield v-number 1 or 2 rests on exhaustive colon-ideal case analysis; it is unclear whether the argument enumerates all minimal generators correctly when m>2, which is load-bearing for the classification claim.

Authors: We thank the referee for highlighting this potential gap in clarity. The colon-ideal case analysis in Section 4 is formulated for general m by considering the generators of J_{K_m,G} as the union of the binomial generators from G and those from the complete graph K_m; the possible colon operations with respect to variables or monomials are classified according to whether they involve vertices in the cut set or not, and this enumeration does not depend on the specific value of m beyond the number of variables contributed by K_m. To make this explicit, we have added a new remark and a short lemma in the revised Section 4 that lists the minimal generators of the relevant colon ideals for arbitrary m (with m=3 worked out in full as an example). The classification statements themselves remain unchanged. revision: yes
Referee: [§5.2] §5.2 (linearity for powers): the claim that v(J_{K_2,G}^k) is linear in k is derived from the Cohen-Macaulay hypothesis via iterated colon ideals; the manuscript must exhibit the precise recurrence or closed-form coefficient to confirm the linearity is not an artifact of the chosen generators.

Authors: We agree that an explicit recurrence strengthens the presentation. Under the Cohen-Macaulay hypothesis, the iterated colon ideals J_{K_2,G}^k : f (for a suitable minimal generator f) yield a constant increment independent of k. In the revised Section 5.2 we now state the precise recurrence v(J_{K_2,G}^{k+1}) = v(J_{K_2,G}^k) + c, where the constant c equals the number of vertices minus the size of a maximum matching in the closed graph G (explicitly computed from the primary decomposition). This immediately gives the closed form v(J_{K_2,G}^k) = v(J_{K_2,G}) + (k-1)c, confirming linearity directly from the colon computations rather than from any special choice of generators. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivations proceed by direct investigation of colon ideals of J_{K_m,G} to obtain explicit formulas for the local v-number (with respect to the empty cut set), followed by classification of graphs yielding v-numbers 1 or 2, generalization of Dey et al. for closed graphs, and further formulas under the Cohen-Macaulay hypothesis that establish linearity in k for powers. These steps rely on standard algebraic operations (colon ideals, primary decompositions, and graph-theoretic hypotheses stated upfront) rather than any self-definitional reduction, fitted-parameter renaming, or load-bearing self-citation chain. The cited prior work of Dey et al. is external and the results remain falsifiable via direct computation on the given ideals, rendering the chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard definition of generalized binomial edge ideals in a polynomial ring and the algebraic properties of colon ideals; no free parameters or invented entities are introduced.

axioms (2)

domain assumption G is a finite connected simple graph
Explicitly stated as the setup for defining J_{K_m,G} and all subsequent results.
standard math Standard properties of colon ideals and v-numbers in commutative algebra hold
Invoked implicitly when deriving the formula from colon ideals.

pith-pipeline@v0.9.0 · 5482 in / 1530 out tokens · 69814 ms · 2026-05-13T23:43:16.428260+00:00 · methodology

discussion (0)

Reference graph

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