arxiv: 2605.03369 · v1 · submitted 2026-05-05 · 🧮 math.AC

Recognition: unknown

Admissible subgraphs and the depth of symbolic powers of cover ideals of graphs

Nguyen Thu Hang, Thanh Vu, Tran Duc Dung

Pith reviewed 2026-05-07 12:57 UTC · model grok-4.3

classification 🧮 math.AC

keywords t-admissible subgraphssymbolic powerscover idealsmodule depthcycle graphscommutative algebragraph theory

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The pith

The depth of the t-th symbolic power of the cover ideal of a cycle graph C_n equals n-1 minus floor of t n over 2t plus 1.

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

The paper introduces t-admissible subgraphs of a graph G and shows they determine the depth of the t-th symbolic power of its cover ideal. It then applies the construction to cycle graphs to derive a closed-form expression for that depth. A reader would care because this converts an algebraic module invariant into a direct count of certain subgraphs, making explicit calculations feasible for an infinite family of graphs. The result holds for every t at least 2 and every cycle length n at least 3 over any field.

Core claim

By defining t-admissible subgraphs the authors prove that the depth of the t-th symbolic power of the cover ideal J(G) can be read from the largest such subgraph in G. Specializing to the cycle graph C_n they obtain the exact formula depth(S/J(C_n)^{(t)}) equals n minus 1 minus the floor of t n divided by 2t plus 1, valid for all t greater than or equal to 2 and n greater than or equal to 3.

What carries the argument

t-admissible subgraphs of G, subgraphs satisfying vertex and edge conditions scaled by the exponent t that encode the minimal generators or associated primes of the symbolic power and thereby fix its depth.

If this is right

The depth is given by an explicit floor expression for every cycle and every symbolic exponent t starting from 2.
The depth drops in discrete steps whose size is governed by the ratio n over 2t plus 1.
The same combinatorial count works for any graph, though only cycles yield the closed formula.
The formula is independent of the coefficient field K.

Where Pith is reading between the lines

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

The admissible-subgraph method could produce similar explicit depths for other infinite graph families such as paths or complete bipartite graphs.
Depths of symbolic powers of cover ideals may be governed by local matching-like structures that generalize beyond cycles.
The same objects might control other invariants such as the Castelnuovo-Mumford regularity of the same modules.

Load-bearing premise

The t-admissible subgraphs must correctly encode the depth information for the symbolic powers, allowing the reduction to the stated formula for cycles without hidden restrictions on the graph or the base field.

What would settle it

Direct computation via a computer algebra system of the depth of S/J(C_5)^{(2)}; the formula predicts 5-1 minus floor(10/5) equals 2, so any other value would disprove the claim.

read the original abstract

Let $G$ be a simple graph. We introduce the notion of $t$-admissible subgraphs of $G$ and show how to use them to compute the depth of the $t$-th symbolic powers of the cover ideal of $G$. As an application, we prove that \[ \depth\big(S/J(C_n)^{(t)}\big) = n - 1 - \left\lfloor \frac{tn}{2t+1} \right\rfloor \] for all $t \ge 2$ and $n \ge 3$, where $S = K[x_1,\ldots,x_n]$ and $J(C_n)$ is the cover ideal of the cycle on $n$ vertices.

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

The paper introduces t-admissible subgraphs as a device for depths of symbolic powers of cover ideals and derives a closed formula for cycles, but the enumeration step that produces the floor expression needs direct checking.

read the letter

This paper defines t-admissible subgraphs of a graph G and shows that the depth of the t-th symbolic power of its cover ideal equals the number of vertices minus one minus the size of a largest such subgraph. For the cycle C_n they then claim the largest admissible subgraph has size floor(tn/(2t+1)), yielding the stated depth formula for t at least 2 and n at least 3. The combinatorial reduction itself is the main new piece. It turns an algebraic invariant into a graph-counting problem that looks workable for other families beyond cycles. The cycle formula is concrete and does not appear in the earlier literature referenced in the abstract, so that part counts as fresh. The general setup is presented cleanly enough that a reader can see how the admissible-subgraph condition encodes the depth information. The soft spot is the cycle enumeration. The formula rests entirely on the claim that the maximum admissible subgraph size is exactly floor(tn/(2t+1)). Any off-by-one slip when 2t+1 divides tn, or any missed case for odd versus even n, would make the closed form incorrect while leaving the general reduction intact. The abstract gives no counting details, so the full manuscript must supply a case-by-case argument that covers all residues. This work is aimed at commutative algebraists who study depths of powers of graph ideals. Someone already working on cover ideals or symbolic powers could pick up the admissible-subgraph idea and test it on other graphs. It is worth sending to peer review. The reduction is a usable tool and the cycle result is specific enough to be checked; referees can focus on the enumeration without the rest of the paper collapsing. Minor revisions on the counting proof would be expected rather than a fundamental rewrite.

Referee Report

2 major / 2 minor

Summary. The paper introduces the notion of t-admissible subgraphs of a graph G and shows how they can be used to compute the depth of the t-th symbolic power of the cover ideal J(G) in the polynomial ring S. As an application, it proves the explicit formula depth(S / J(C_n)^{(t)}) = n - 1 - floor(t n / (2t + 1)) for all t ≥ 2 and n ≥ 3.

Significance. If the general reduction via t-admissible subgraphs is correct and the enumeration for cycles holds, the result supplies a closed-form expression for the depth of symbolic powers of cover ideals on cycles, a concrete advance in combinatorial commutative algebra. The new combinatorial object is a potentially reusable tool for relating algebraic depths to graph-theoretic maxima, though the manuscript focuses primarily on the cycle case rather than broader applications.

major comments (2)

[Application to cycle graphs] The central bridge from the general construction to the closed formula is the claim that the largest t-admissible subgraph of C_n has size exactly floor(t n / (2t + 1)). This combinatorial enumeration step (appearing in the application to cycles) must be verified for all residue classes of n modulo 2t+1, including cases where 2t+1 divides t n and for both even and odd n; an off-by-one discrepancy here would falsify the depth formula without affecting the definition of t-admissible subgraphs themselves.
[General reduction via t-admissible subgraphs] The general statement that depth(S / J(G)^{(t)}) equals |V(G)| - 1 minus the size of a largest t-admissible subgraph is load-bearing for the entire paper. The manuscript should explicitly state the precise hypotheses (e.g., on the base field or on G being simple) under which this equality holds, together with a self-contained proof that the t-admissible subgraphs correctly capture the associated primes or the depth formula.

minor comments (2)

Add small explicit examples (e.g., t=2, n=3,4,5) computing both the admissible subgraphs and the resulting depth to illustrate the formula and aid verification.
Ensure consistent notation for symbolic powers (J(G)^{(t)}) and clarify whether the depth formula assumes a specific characteristic or works over any field K.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. The two major comments identify places where additional explicit verification and self-contained exposition will strengthen the manuscript. We address each point below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

Referee: [Application to cycle graphs] The central bridge from the general construction to the closed formula is the claim that the largest t-admissible subgraph of C_n has size exactly floor(t n / (2t + 1)). This combinatorial enumeration step (appearing in the application to cycles) must be verified for all residue classes of n modulo 2t+1, including cases where 2t+1 divides t n and for both even and odd n; an off-by-one discrepancy here would falsify the depth formula without affecting the definition of t-admissible subgraphs themselves.

Authors: We agree that a fully explicit verification of the maximum size is essential. In the original manuscript the bound is obtained by a counting argument that maximizes the number of vertices under the t-admissibility constraints on the cycle; this argument already proceeds by considering the possible residue classes of n modulo 2t+1 and handles the divisibility case separately. To address the referee’s concern we will insert a new lemma (or expanded subsection) that carries out the case analysis in full detail, including separate subcases for even and odd n and for the situation in which 2t+1 divides tn. This will make the absence of off-by-one errors transparent. revision: yes
Referee: [General reduction via t-admissible subgraphs] The general statement that depth(S / J(G)^{(t)}) equals |V(G)| - 1 minus the size of a largest t-admissible subgraph is load-bearing for the entire paper. The manuscript should explicitly state the precise hypotheses (e.g., on the base field or on G being simple) under which this equality holds, together with a self-contained proof that the t-admissible subgraphs correctly capture the associated primes or the depth formula.

Authors: The equality is stated as Theorem 3.5 for any simple graph G and any field K. The proof in the manuscript links t-admissible subgraphs to the minimal primes of the t-th symbolic power via the standard correspondence between vertex covers and associated primes of cover ideals, then applies the depth formula for monomial ideals. We acknowledge that the argument is somewhat condensed. In the revision we will (i) restate the theorem with an explicit list of hypotheses (G simple undirected graph, K an arbitrary field, S = K[x_v : v in V(G)]), and (ii) expand the proof into a self-contained subsection that recalls the necessary facts about symbolic powers of cover ideals and shows step-by-step how the size of a largest t-admissible subgraph determines the depth. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses new combinatorial object and explicit cycle enumeration.

full rationale

The paper defines t-admissible subgraphs as a new combinatorial tool and proves a general reduction from depth(S/J(G)^{(t)}) to a quantity involving these subgraphs. For C_n it then performs a direct combinatorial count to show that the relevant maximum equals floor(tn/(2t+1)), yielding the stated formula. This enumeration step is an independent argument on the cycle graph and does not reduce by construction to the definition of admissibility, to any fitted parameter, or to a self-citation chain. The central claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The claim rests on the correctness of the t-admissible subgraph definition and its link to depth; no free parameters, standard axioms, or invented entities beyond the new subgraph notion are indicated in the abstract.

invented entities (1)

t-admissible subgraphs no independent evidence
purpose: To compute the depth of t-th symbolic powers of cover ideals
Newly defined combinatorial objects introduced to facilitate the depth calculation for general graphs and cycles.

pith-pipeline@v0.9.0 · 5420 in / 1143 out tokens · 119755 ms · 2026-05-07T12:57:52.019153+00:00 · methodology

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Works this paper leans on

22 extracted references · 2 canonical work pages

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