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

Nguyen Thu Hang; Thanh Vu; Tran Duc Dung

arxiv: 2605.03369 · v2 · pith:N3TLD6PRnew · submitted 2026-05-05 · 🧮 math.AC

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

Tran Duc Dung , Nguyen Thu Hang , Thanh Vu This is my paper

Pith reviewed 2026-05-21 00:43 UTC · model grok-4.3

classification 🧮 math.AC

keywords cover idealsymbolic powerdepthcycle graphadmissible subgraphgraph idealcommutative algebra

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The pith

The depth of the t-th symbolic power of the cover ideal of cycle C_n is n - 1 - floor(t n / (2 t + 1)) for t at least 2.

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

This paper introduces the concept of t-admissible subgraphs of a graph G. It shows that these subgraphs can be used to compute the depth of the t-th symbolic power of the cover ideal of G. The authors apply this technique to cycle graphs to obtain an explicit formula for the depth. This matters for determining the homological behavior of these ideals in the polynomial ring over a field.

Core claim

The central claim is that t-admissible subgraphs allow computation of the depth of symbolic powers of cover ideals, and for the cycle on n vertices the depth equals n minus 1 minus the floor of t times n divided by 2t plus 1.

What carries the argument

t-admissible subgraphs of G, which encode the combinatorial data to find the depth of J(G) to the power t symbolically.

If this is right

The depth formula applies to all cycles with at least 3 vertices and all symbolic powers starting from t=2.
The method provides a general way to calculate depths for cover ideals of arbitrary simple graphs.
These depths give information on the minimal number of generators in the resolution of the symbolic powers.

Where Pith is reading between the lines

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

Similar admissible notions could be developed for other graph ideals like the edge ideal.
The formula suggests that the depth decreases roughly by n/(2t+1) at each step of t, which might relate to asymptotic behavior.
This combinatorial method may help in studying symbolic powers in other algebraic settings involving graphs.

Load-bearing premise

The newly defined t-admissible subgraphs correctly determine the depth of the symbolic powers of the cover ideal.

What would settle it

Direct computation of the depth for the cycle C_5 with t=2, where the formula gives 5-1-floor(10/5)=4-2=2, and checking if the actual depth matches this value.

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

This paper defines t-admissible subgraphs and uses them to prove an explicit depth formula for the symbolic powers of cover ideals on cycles.

read the letter

The main takeaway is that this paper defines a new combinatorial object called t-admissible subgraphs and uses it to compute the depth of symbolic powers of cover ideals, leading to an explicit formula for cycle graphs. They prove that the depth of the t-th symbolic power of the cover ideal of C_n is n - 1 - floor(t n / (2 t + 1)). This gives a closed form that was not known before. What the paper does well is to offer a concrete way to find these depths without relying solely on algebraic properties like associated primes. By introducing t-admissible subgraphs, they create a bridge between the graph structure and the algebraic invariant. The application to cycles is clean and gives a simple expression in terms of n and t. The soft spot, if there is one, lies in the verification for the cycle. The general theorem expresses the depth in terms of the largest value associated to t-admissible subgraphs. For the formula to hold, they must show that this largest value is precisely the floor expression. Their argument probably involves selecting vertices or edges in a periodic fashion around the cycle. One should check whether this works uniformly when n is odd or even, or when t is large relative to n. But if the counting is accurate, there is no issue. This kind of result is useful for researchers studying monomial ideals and their depths in the context of graphs. It provides a specific computation that can serve as a test case or building block for more general graphs. A reader who follows papers on cover ideals, symbolic powers, and graph-theoretic algebra would get value from this. It is not a broad theory but a targeted advance. The paper shows clear thinking in setting up the new definition and applying it. It deserves a serious referee. I would recommend that an editor send it out for peer review.

Referee Report

2 major / 2 minor

Summary. The paper introduces the notion of t-admissible subgraphs of a simple graph G and shows how they determine the depth of the t-th symbolic power of the cover ideal J(G). As an application, it proves the closed-form expression depth(S/J(C_n)^{(t)}) = n-1 - floor(tn/(2t+1)) for all t≥2 and n≥3, where S=K[x1,...,xn].

Significance. If the central identification holds, the work supplies a combinatorial criterion for depths of symbolic powers of cover ideals, an invariant of interest in commutative algebra and algebraic combinatorics. The explicit formula for cycles yields a concrete, testable prediction and illustrates the method on a family where direct computation is feasible.

major comments (2)

[Application to cycles] Application to cycles (the formula stated in the abstract): the claim that the maximum value of the parameter attached to t-admissible subgraphs of C_n equals exactly floor(tn/(2t+1)) is load-bearing for the depth formula. The manuscript must supply a self-contained combinatorial argument (induction or counting) that accounts for the cyclic wrapping and the t-multiplicity; any mismatch in the extremal count would falsify the stated depth.
[Main results] General method (the relation between depth and t-admissible subgraphs): the equality depth(S/J(G)^{(t)}) = n-1 minus the maximum parameter over t-admissible subgraphs must be proved directly from the definition rather than asserted; the proof should not rely on the cycle case to establish the general link.

minor comments (2)

[Abstract] The abstract would benefit from a one-sentence statement of the general depth formula before the cycle application.
[Introduction] Notation for the ring S and the variables x1,...,xn should be introduced explicitly in the first section rather than assumed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address the major comments point by point below and have made revisions to strengthen the proofs as requested.

read point-by-point responses

Referee: [Application to cycles] Application to cycles (the formula stated in the abstract): the claim that the maximum value of the parameter attached to t-admissible subgraphs of C_n equals exactly floor(tn/(2t+1)) is load-bearing for the depth formula. The manuscript must supply a self-contained combinatorial argument (induction or counting) that accounts for the cyclic wrapping and the t-multiplicity; any mismatch in the extremal count would falsify the stated depth.

Authors: We agree that providing a self-contained combinatorial argument is crucial for validating the depth formula for cycles. In the revised version of the manuscript, we will incorporate a detailed proof by induction on n that carefully handles the cyclic wrapping and the t-multiplicity in the definition of t-admissible subgraphs. This will confirm that the maximum parameter is exactly floor(tn/(2t+1)). revision: yes
Referee: [Main results] General method (the relation between depth and t-admissible subgraphs): the equality depth(S/J(G)^{(t)}) = n-1 minus the maximum parameter over t-admissible subgraphs must be proved directly from the definition rather than asserted; the proof should not rely on the cycle case to establish the general link.

Authors: We acknowledge the need for a direct proof of the general relation. We will revise the manuscript to include a standalone proof of the equality depth(S/J(G)^{(t)}) = n-1 - max parameter, derived directly from the definitions of symbolic powers, cover ideals, and t-admissible subgraphs, using standard techniques from commutative algebra such as the primary decomposition of monomial ideals. This proof will be independent of the cycle application. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new combinatorial notion yields independent depth formula

full rationale

The paper introduces the definition of t-admissible subgraphs as a new combinatorial object and proves a general theorem equating depth(S/J(G)^{(t)}) to n-1 minus a maximum parameter over these subgraphs. For the cycle C_n the authors then perform a separate combinatorial count to identify that maximum as floor(tn/(2t+1)). This count is presented as an explicit enumeration on the cycle (accounting for wrapping and multiplicity t), not as a redefinition or tautological fit of the target depth expression. No self-citation chain, ansatz smuggling, or fitted-input renaming is indicated; the derivation remains self-contained once the admissible-subgraph parameter is accepted as independently defined.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the correctness of the newly defined t-admissible subgraphs and on standard facts about depth and symbolic powers; no numerical parameters are fitted.

axioms (1)

standard math Depth of a module over a polynomial ring satisfies the usual properties from commutative algebra (e.g., relation to associated primes and regular sequences).
Background assumed throughout the field; invoked implicitly when depth is computed.

invented entities (1)

t-admissible subgraph no independent evidence
purpose: To compute the depth of the t-th symbolic power of the cover ideal
Newly introduced combinatorial object whose maximal size or structure is claimed to determine the depth.

pith-pipeline@v0.9.0 · 5651 in / 1307 out tokens · 47051 ms · 2026-05-21T00:43:51.795796+00:00 · methodology

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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.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Critical subgraphs and the regularity of symbolic powers of cover ideals of graphs
math.AC 2026-05 unverdicted novelty 5.0

A method based on t-admissible subgraphs determines the regularity of the t-th symbolic power of cover ideals of graphs, applied to bipartite unicyclic graphs.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

Balanescu and M

S. Balanescu and M. Cimpoeas,Depth and Stanley depth of powers of the path ideal of a path graph, U.P.B. Sci. Bull., Series A86 (4)(2024), 65–76. 2

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J. A. Bondy, U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics244, Springer, New York 2008. 3

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Beyarslan, H.T

S. Beyarslan, H.T. Ha and T.N. Trung,Regularity of powers of forests and cycles, J. Algebraic Combin.42(2015), 1077–1095. 5

work page 2015
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M.P. Binh, N. T. Hang, T. T. Hien, T. N. Trung,stability of cover ideals, J. Algebraic Combin. 63(2026), 3. 2, 9

work page 2026
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T.D. Dung, N. T. Hang, P. H. Nam, N. T. T. Tam,The depth function of powers of cover ideals of path graphs, arXiv:2605.03347. 2

work page internal anchor Pith review Pith/arXiv arXiv
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work page 2024
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N. C. Minh, T. N. Trung, and T. Vu,Depth of powers of edge ideals of cycles and starlike trees, to appear in Rocky M. J. Math. arXiv:2308.00874. 2

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T. N. Trung,Stability of depths of powers of edge ideals, J. Algebra452(2016), 157–187. 1 Thai Nguyen University of Sciences, Phan Dinh phung W ard, Thai Nguyen, Vietnam Email address:dungtd@tnus.edu.vn Thai Nguyen University of Sciences, Phan Dinh phung W ard, Thai Nguyen, Vietnam Email address:hangnt@tnus.edu.vn Institute of Mathematics, V AST, 18 Hoang...

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[1] [1]

Balanescu and M

S. Balanescu and M. Cimpoeas,Depth and Stanley depth of powers of the path ideal of a path graph, U.P.B. Sci. Bull., Series A86 (4)(2024), 65–76. 2

work page 2024

[2] [2]

J. A. Bondy, U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics244, Springer, New York 2008. 3

work page 2008

[3] [3]

Beyarslan, H.T

S. Beyarslan, H.T. Ha and T.N. Trung,Regularity of powers of forests and cycles, J. Algebraic Combin.42(2015), 1077–1095. 5

work page 2015

[4] [4]

M.P. Binh, N. T. Hang, T. T. Hien, T. N. Trung,stability of cover ideals, J. Algebraic Combin. 63(2026), 3. 2, 9

work page 2026

[5] [5]

Brodmann,The asymptotic nature of the analytic spread, Math

M. Brodmann,The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc.86(1979), 35–39. 1

work page 1979

[6] [6]

Constantinescu and M

A. Constantinescu and M. Varbaro,Koszulness, Krull dimension, and other properties of graph- related algebras, J. Algebr. Comb.,34(2011), 375–400. 9

work page 2011

[7] [7]

T.D. Dung, N. T. Hang, P. H. Nam, N. T. T. Tam,The depth function of powers of cover ideals of path graphs, arXiv:2605.03347. 2

work page internal anchor Pith review Pith/arXiv arXiv

[8] [8]

C. A. Francisco, H. T. Ha, and A. Van Tuyl,Associated primes of monomial ideals and odd holes in graphs, J Algebr Comb.32(2010), 287–301. 9

work page 2010

[9] [9]

Herzog, T

J. Herzog, T. Hibi and N. V. Trung,Symbolic powers of monomial ideals and vertex cover algebras, Adv. Math.210(1) (2007), 304–322. 2

work page 2007

[10] [10]

Herzog and T

J. Herzog and T. Hibi,Monomial ideals, Graduate Texts in Mathematics260, Springer, New York 2010. 2

work page 2010

[11] [11]

T. T. Hien, N. T. Hang, and T. Vu,Depth of powers of edge ideals of Cohen-Macaulay trees, Communications in Algebra52(2024), 5049–5060. 2

work page 2024

[12] [12]

L. T. Hoa, K. Kimura, N. Terai and T. N. Trung,Stability of depths of symbolic powers of Stanley-Reisner ideals, J. Algebra,473(2017), 307-323. 2, 9

work page 2017

[13] [13]

L. T. Hoa and T. N. Trung,Partial Castelnuovo-Mumford regularities of sums and intersections of powers of monomial ideals, Math. Proc. Cambridge Philos. Soc.149(2010), 1–18. 1 10 T. D. DUNG, N.T. HANG, AND THANH VU

work page 2010

[14] [14]

Hochster, Cohen - Macaulay rings, combinatorics, and simlpicial complexes, inB

M. Hochster, Cohen - Macaulay rings, combinatorics, and simlpicial complexes, inB. R. Mc- Donald and R. A. Morris (eds), Ring theory II, Lect. Notes in Pure and Appl. Math. 26, M. Dekker, 1977, 171–223. 2

work page 1977

[15] [15]

Jacques,Betti numbers of graph ideals, Ph.D

S. Jacques,Betti numbers of graph ideals, Ph.D. Thesis, University of Sheffield, 2004. 3

work page 2004

[16] [16]

H. M. Lam, N. V. Trung, and T. N. Trung,A general formula for the index of depth stability of edge ideals, Trans. Amer. Math. Soc.377(2024), 8633–8657. 1

work page 2024

[17] [17]

N. C. Minh, T. N. Trung, and T. Vu,Depth of powers of edge ideals of cycles and starlike trees, to appear in Rocky M. J. Math. arXiv:2308.00874. 2

work page arXiv

[18] [18]

N. C. Minh, T. N. Trung, and T. Vu,Stable value of depth of symbolic powers of edge ideals of graphs, Pacific Journal of Mathematics329(2024), 147–164. 1

work page 2024

[19] [19]

Simis, W.V

A. Simis, W.V. Vasconcelos, and R. H. Villarreal,On the ideal theory of graphs, J. Algebra167 (1994), 389–416. 2

work page 1994

[20] [20]

N. Terai,Alexander duality theorem and Stanley-Reisner rings, Surikaisekikenkyusho Kokyuroku 1078, 174–184, Free resolutions of coordinate rings of projective varieties and related topics (Kyoto, 1998). 5

work page 1998

[21] [21]

T. N. Trung,Stability of depths of powers of edge ideals, J. Algebra452(2016), 157–187. 1 Thai Nguyen University of Sciences, Phan Dinh phung W ard, Thai Nguyen, Vietnam Email address:dungtd@tnus.edu.vn Thai Nguyen University of Sciences, Phan Dinh phung W ard, Thai Nguyen, Vietnam Email address:hangnt@tnus.edu.vn Institute of Mathematics, V AST, 18 Hoang...

work page 2016