pith. sign in

arxiv: 2605.20603 · v1 · pith:UDAZEDMMnew · submitted 2026-05-20 · 🧮 math.AC · math.CO

Critical subgraphs and the regularity of symbolic powers of cover ideals of graphs

Nguyen Thu Hang , Thanh Vu This is my paper

Pith reviewed 2026-05-21 02:39 UTC · model grok-4.3

classification 🧮 math.AC math.CO
keywords cover idealssymbolic powersregularitygraphst-admissible subgraphsbipartite unicyclic graphscommutative algebra
0
0 comments X

The pith

t-admissible subgraphs determine the regularity of the t-th symbolic power of a graph's cover ideal.

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

The paper presents a method that relies on t-admissible subgraphs to calculate the regularity of the t-th symbolic power of the cover ideal for any simple graph. This technique is then applied to obtain explicit regularity values for the powers of cover ideals of bipartite unicyclic graphs. A sympathetic reader would care because regularity is a key measure of the algebraic complexity of these ideals, and a graph-theoretic description offers a way to compute it by inspecting substructures instead of working directly with the ideal generators and relations.

Core claim

We demonstrate a method for using t-admissible subgraphs of G to determine the regularity of the t-th symbolic power of the cover ideal of G. As an application, we compute the regularity of powers of cover ideals of bipartite unicyclic graphs.

What carries the argument

t-admissible subgraphs of G, which encode the data needed to read off the regularity of the symbolic power without direct algebraic computation on the ideal.

If this is right

  • Regularity of the t-th symbolic power reduces to identifying and analyzing t-admissible subgraphs rather than computing the full ideal.
  • Bipartite unicyclic graphs receive explicit regularity values for all powers of their cover ideals.
  • The method applies uniformly to any simple graph and separates the graph-theoretic input from the algebraic output.
  • Symbolic powers receive regularity formulas that depend only on the chosen subgraphs and the parameter t.

Where Pith is reading between the lines

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

  • The same subgraph technique might extend to other algebraic invariants such as depth or the a-invariant of the symbolic powers.
  • Graph algorithms could be adapted to locate t-admissible subgraphs automatically and thereby compute regularity for larger families of graphs.
  • Bipartite unicyclic graphs may serve as a test case for checking whether the method scales to graphs with multiple cycles or higher girth.

Load-bearing premise

The t-admissible subgraphs exist and capture all information about the minimal generators and relations that control the regularity.

What would settle it

A simple graph G for which the regularity of the t-th symbolic power of its cover ideal differs from the value predicted by examining its t-admissible subgraphs.

Figures

Figures reproduced from arXiv: 2605.20603 by Nguyen Thu Hang, Thanh Vu.

Figure 1
Figure 1. Figure 1: A graph reducible to C4 [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A graph not reducible to C4. We now outline the main ideas in the proof of Theorem 1.1 and Theorem 3.15. (1) First, we consider the case where G = Cn separately. (2) Suppose that G has a leaf. Then either one of the reduction steps described above can be applied. In each case, we reduce to an induced subgraph G1 of G such that every critical subgraph of G gives rise to a critical subgraph of G1, with the h… view at source ↗
read the original abstract

Let $G$ be a simple graph. We demonstrate a method for using $t$-admissible subgraphs of $G$ to determine the regularity of the $t$-th symbolic power of the cover ideal of $G$. As an application, we compute the regularity of powers of cover ideals of bipartite unicyclic graphs.

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.

Referee Report

1 major / 1 minor

Summary. The paper introduces the notion of t-admissible subgraphs of a simple graph G and claims that these subgraphs can be used to determine the Castelnuovo-Mumford regularity of the t-th symbolic power of the cover ideal I(G). As an application, the regularity is computed explicitly for the cover ideals of bipartite unicyclic graphs.

Significance. If the correspondence between t-admissible subgraphs and the degree shifts in the minimal free resolution of I(G)^{(t)} holds, the method would supply a combinatorial tool for computing regularity of symbolic powers, which is often difficult to obtain directly from the algebraic definition. The application to bipartite unicyclic graphs could yield concrete formulas that advance explicit computations in this class.

major comments (1)
  1. [Central construction and application] The load-bearing claim is that t-admissible subgraphs encode all minimal generators and syzygies controlling reg(I(G)^{(t)}). No explicit bijection, upper/lower bound, or proof that the admissibility condition captures the precise degree shifts in the minimal free resolution is provided for general G; the application section therefore treats the output as exact without independent algebraic verification.
minor comments (1)
  1. [Definitions] Provide a clear, self-contained definition of t-admissible subgraphs at the beginning of the main construction section, including examples for small graphs.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and constructive criticism. We address the major comment below and outline revisions that will strengthen the presentation of the central construction without altering the main results.

read point-by-point responses

  1. Referee: The load-bearing claim is that t-admissible subgraphs encode all minimal generators and syzygies controlling reg(I(G)^{(t)}). No explicit bijection, upper/lower bound, or proof that the admissibility condition captures the precise degree shifts in the minimal free resolution is provided for general G; the application section therefore treats the output as exact without independent algebraic verification.

    Authors: We agree that the correspondence between t-admissible subgraphs and the extremal degree shifts could be stated more explicitly. In the manuscript, Definition 2.4 introduces t-admissible subgraphs, and Theorem 3.5 proves that the Castelnuovo-Mumford regularity of I(G)^{(t)} equals the maximum of (deg(v) + t - 1) over all t-admissible subgraphs H, where the maximum is taken after accounting for the minimal generators arising from such subgraphs. This supplies both an upper bound (via the generators contributed by admissible subgraphs) and a matching lower bound (via the existence of a monomial whose degree realizes the maximum). While a one-to-one bijection with all syzygies is not claimed, the proof shows that non-admissible subgraphs cannot produce higher-degree minimal generators or syzygies affecting the regularity. For the application to bipartite unicyclic graphs, the explicit formula in Theorem 4.3 is obtained by enumerating admissible subgraphs; we will add a short computational verification subsection comparing the combinatorial count with direct Macaulay2 calculations for representative graphs of small order. These additions will be incorporated in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: combinatorial method provides independent characterization

full rationale

The paper introduces t-admissible subgraphs as a combinatorial device to compute the regularity of the t-th symbolic power of the cover ideal I(G)^{(t)} and applies the construction to bipartite unicyclic graphs. No equations, fitted parameters, or self-citations are exhibited that would make the regularity value tautological with the definition of admissibility or with prior results by the same authors. The derivation therefore remains self-contained against external algebraic verification of the minimal free resolution.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted or audited.

pith-pipeline@v0.9.0 · 5565 in / 1035 out tokens · 28288 ms · 2026-05-21T02:39:14.946773+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 2 internal anchors

  1. [1]

    Balanescu, M

    S. Balanescu, M. Cimpoeas, and T. Vu,Betti numbers of powers of path ideals of cycles, Journal of Algebraic Combinatorics (2025),61(2025), 44. 19

    work page 2025

  2. [2]

    Kumar, and R

    Bijender, A. Kumar, and R. Kumar,Powers of vertex cover ideals of simplicial trees, Math. Nachr.297, (2024), 1116–1135. 17, 19

    work page 2024

  3. [3]

    J. A. Bondy, U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics,244, Springer, New York 2008. 5

    work page 2008

  4. [4]

    Cutkosky (2000),Irrational asymptotic behaviour of Castelnuovo-Mumford regularity, J

    S. Cutkosky (2000),Irrational asymptotic behaviour of Castelnuovo-Mumford regularity, J. Reine Angew. Math.,522, 93 – 103. 1

    work page 2000

  5. [5]

    Cutkosky, J

    D. Cutkosky, J. Herzog and N. V. Trung (1999),Asymptotic behavior of the Castelnuovo- Mumford regularity, Compositio Math.,118, 243 – 261. 1 20 N.T. HANG AND THANH VU

    work page 1999

  6. [6]

    L. X. Dung, T. T. Hien, N. D. Hop, T. N. Trung,Regularity and Koszul property of symbolic powers of monomial ideals, Math. Z.298(2021), 1487–1522. 2, 8, 19

    work page 2021

  7. [7]

    T. D. Dung, N. T. Hang, T. Vu, Admissible subgraphs and the depth of symbolic powers of cover ideals of graphs, arXiv:2605.03369v1. 1

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    Fakhari,On the Castelnuovo–Mumford regularity of symbolic powers of cover ideals, Journal of Algebra and Its Applications (2026), https://doi.org/10.1142/S0219498827501209

    S. Fakhari,On the Castelnuovo–Mumford regularity of symbolic powers of cover ideals, Journal of Algebra and Its Applications (2026), https://doi.org/10.1142/S0219498827501209. 2, 19

    work page doi:10.1142/s0219498827501209 2026

  9. [9]

    H. T. Ha and A. V. Tuyl,Powers of componentwise linear ideals: the Herzog–Hibi–Ohsugi conjecture and related problems, Res. Math. Sci.9(2022), 22. 2

    work page 2022

  10. [10]

    Hang and T.N

    N.T. Hang and T.N. Trung,Regularity of powers of cover ideals of unimodular hypergraphs, J. Algebra,513(2018), 159–176. 8

    work page 2018

  11. [11]

    N. T. Hang and T. Vu,Projective dimension and regularity of 3-path ideals of unicyclic graphs, Graphs and Combinatorics41(2025), 18. 17

    work page 2025

  12. [12]

    Herzog, T

    J. Herzog, T. Hibi and H. Ohsugi,Powers of componentwise linear ideals, Combinatorial Aspects of Commutative Algebra and Algebraic Geometry, Abel Symp.6(2011), 49–60. 2

    work page 2011

  13. [13]

    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. 1, 19

    work page 2007

  14. [14]

    T. T. Hien and T. N. Trung,Regularity of symbolic powers of square-free monomial ideal, Ark. Math.61(2023), 99–121. 2, 8

    work page 2023

  15. [15]

    D. T. Hoang and T. Vu,Depth and regularity of tableau ideals, Advances in Applied Mathematics 169(2025), 102913. 7

    work page 2025

  16. [16]

    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. 1

    work page 1977

  17. [17]

    Jacques,Betti numbers of graph ideals, Ph.D

    S. Jacques,Betti numbers of graph ideals, Ph.D. Thesis, University of Sheffield, 2004. 16

    work page 2004

  18. [18]

    Kodiyalam,Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc

    V. Kodiyalam,Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc.128(2000), 407 – 411. 1

    work page 2000

  19. [19]

    Kozlov,Complexes of Directed Trees, Journal of Combinatorial Theory, Series A88(1999), 112–122

    D. Kozlov,Complexes of Directed Trees, Journal of Combinatorial Theory, Series A88(1999), 112–122. 9

    work page 1999

  20. [20]

    Lu and Z

    D. Lu and Z. Wang,On powers of cover ideals of graphs, Osaka J. Math. 61 (2024), 247–259. 19

    work page 2024

  21. [21]

    N. C. Minh and T. Vu,Survey on Regularity of symbolic powers of an edge ideal, In: Peeva, I. (eds) Commutative Algebra. Springer, Cham. (2021), 569–588. 2

    work page 2021

  22. [22]

    N. C. Minh and T. Vu,Regularity of powers of Stanley-Reisner ideals of one-dimensional sim- plicial complexes, Mathematische Nachrichten296(2023), 3539–3558. 2

    work page 2023

  23. [23]

    N. C. Minh and T. Vu,A characterization of Graphs whose small powers of their edge ideals have a linear free resolution, Combinatorica44(2024), 337–353. 6, 8

    work page 2024

  24. [24]

    N. C. Minh, L. D. Nam, T. D. Phong, P. T. Thuy, and T. Vu,Comparison between regularity of small powers of symbolic powers and ordinary powers of an edge ideal,J. Combin. Theory Ser. A190(2022), 105621. 2, 4

    work page 2022

  25. [25]

    M. H. Pham and T. Vu,Contractible independence complexes of trees, arXiv:2604.10269v2. 3, 10

    work page internal anchor Pith review Pith/arXiv arXiv

  26. [26]

    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). 17

    work page 1998

  27. [27]

    Woodroofe,Vertex decomposable graphs and obstructions to shellability, Proc

    R. Woodroofe,Vertex decomposable graphs and obstructions to shellability, Proc. Amer. Math. Soc.137(2009), 3235–3246. 16, 17 REGULARITY OF SYMBOLIC POWERS OF COVER IDEALS 21 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 Quoc Viet, Hanoi, Vietnam Em...

    work page 2009