Regularity of Edge Ideals Via Suspension

Arindam Banerjee; Eran Nevo

REVIEW 2 major objections 6 minor 26 references

Regularity of Edge Ideals Via Suspension

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves sharp linear bounds on the regularity of powers of bipartite edge ideals, with a square bound valid for all graphs.

desk verdict Sharp bound for bipartite edge ideals and a new s=2 case; the topological proof is compressed but the mathematics looks right. read the letter →

arxiv 1908.03115 v2 pith:OCVM5XOM submitted 2019-08-08 math.AC math.CO

Arindam Banerjee , Eran Nevo This is my paper · ORCID

classification math.ACmath.CO MSC 13D0205E4013F55

keywords Castelnuovo-MumfordregularityedgeidealspowersofbipartitegraphssimplicialsuspensionMayer-Vietorissequencecoloncombinatorialcommutativealgebra

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 studies how the Castelnuovo-Mumford regularity of an edge ideal grows when the ideal is raised to powers. It proves that for every finite simple graph $G$, $\operatorname{reg}(I(G)^2) \le \operatorname{reg}(I(G))+2$, and that when $G$ is bipartite, $\operatorname{reg}(I(G)^s) \le 2s+\operatorname{reg}(I(G))-2$ for all $s\ge2$. The bound is best possible: complete bipartite graphs attain equality. The square case is proved by a topological argument built on suspension of simplicial complexes, and the higher-power case follows by passing through colon ideals and short exact sequences. The result confirms the conjectured upper bound for every bipartite graph and supports the broader conjecture that the same inequality holds for all graphs.

What carries the argument

The load-bearing object is the suspension of a simplicial complex, $\Sigma_{a,b}\Delta = \Delta * \{\{a\},\{b\},\varnothing\}$, whose geometric realization is the topological suspension. Theorem 3.1 compares $\Delta=\operatorname{cl}(G^c)$ with the clique complex $\Delta'$ of the graph $G'$ obtained from $G$ by connecting every neighbor of $a$ to every neighbor of $b$, and proves $\operatorname{reg}(\Delta')\le \operatorname{reg}(\Delta)$. This topological inequality controls the colon ideal $(I(G)^2:ab)$, and an algebraic induction based on short exact sequences and Theorem 2.3 carries the bound from squares to all powers.

What would settle it

Take any graph $G$, an edge $ab$, and the induced subcomplexes $A$, $B$, $C$ as in Theorem 3.1; choose $W$ with $\widetilde H_\ell(\Delta'[W])\ne 0$ and set $W_C=A\cap B\cap W$. Compute $\widetilde H_\ell(\Delta[\{a,b\}\cup W_C])$ and compare it with $\widetilde H_\ell(\Sigma_{a,b}(\Delta'[W_C]))$. A mismatch for any one such triple would invalidate Claim One and the proof of the base case $\operatorname{reg}(I(G)^2:ab)\le \operatorname{reg}(I(G))$.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1. For any finite simple graph $G$ with edge ideal $I(G)$, squaring raises regularity by at most $2$: $\operatorname{reg}(I(G)^2) \le \operatorname{reg}(I(G))+2$. If $G$ is bipartite, then for every $s\ge2$, $\operatorname{reg}(I(G)^s) \le 2s+\operatorname{reg}(I(G))-2$. This second inequality is the conjectured universal bound specialized to bipartite graphs, and it is sharp because complete bipartite graphs satisfy $\operatorname{reg}(I(G)^s)=2s$. The proof of the square case is topological: an auxiliary clique complex $\Delta'$ built from $\Delta=\operatorname{cl}(G^c)$ by joining the neighbors of an edge's endpoints is shown to have regularity no larger than $\Delta$ (Theorem 3.1). The passage from squares to all powers uses colon ideals, with the bipartite structure guaranteeing that the relevant colons remain edge ideals on the same bipartition.

Load-bearing premise

The load-bearing premise is that, in Claim One of Theorem 3.1, the subcomplex on the vertex set $\{a,b\}\cup W_C$ is exactly the suspension $\Sigma_{a,b}(\Delta'[W_C])$; this equality is asserted without proof, and the vanishing of the homology that carries the base case depends on it.

Editorial extensions

If this is right

For bipartite graphs, the regularity sequence of powers satisfies $\operatorname{reg}(I(G)^s) \le 2s+\operatorname{reg}(I(G))-2$, matching the conjectured universal bound.
Complete bipartite graphs attain equality, so the bound cannot be improved to a smaller intercept for bipartite graphs purely in terms of $\operatorname{reg}(I(G))$.
The theorem verifies the longstanding inequality $b(I(G))\le \operatorname{reg}(I(G))-2$ for all bipartite graphs, where $b(I(G))$ is the eventual intercept of the linear regularity sequence.
For every finite simple graph, not only bipartite ones, squaring the edge ideal raises regularity by at most $2$, giving the base step needed to test the general conjecture for higher powers.

Reading between the lines

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

Editorial: The square bound $\operatorname{reg}(I(G)^2)\le \operatorname{reg}(I(G))+2$ is proved without the bipartite assumption, so if the gap noted below is repaired, the topological suspension strategy may extend the bound to higher powers for classes of graphs beyond bipartite ones.
Editorial: The iteration from squares to all powers depends on the bipartite colon property; finding an analogue of that property for other graph classes would immediately yield the same sharp bound there.
Editorial: The paper's closing example of a flag-no-square dunce-hat triangulation gives a concrete computational experiment: evaluating $\operatorname{reg}(I(G)^2)$ for that graph would test the related conjecture discussed in Section 4, a computation the authors leave open.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

First thing to know: this paper proves the conjectured bound reg(I(G)^s) ≤ 2s + reg I(G) − 2 for all bipartite graphs, and proves the s=2 case for all graphs. The bound is sharp, and it improves on the earlier bound by Jayanthan–Narayanan–Selvaraja that involved the cochordal number. That is a genuine advance, not a routine extension.

What is new is the suspension argument in Theorem 3.1, which gives reg(∆') ≤ reg(∆) where ∆' comes from the graph operation adding edges between neighborhoods of an edge. Using that, the algebraic part for s>2 is a clean induction via colon ideals. The proof for s=2 is topological (Hochster's formula plus Mayer–Vietoris), which is the part that needs careful reading.

The soft spot is exactly where the reader flagged: Claim One in Theorem 3.1. The equality H_l(Σ_{a,b}(∆'[WC])) = H_l(∆[{a,b}∪WC]) is stated in one line. It is true: on WC the complexes ∆ and ∆' agree, and the suspension join with the two vertices a,b gives the induced subcomplex. But the paper does not spell out why, and a non-specialist could get stuck. The same for the subsequent vanishing H_{l−1}(∆'[WC]) = 0. This is an exposition problem, not a mathematical gap, as far as I can tell. I would ask the authors to expand this claim in the next version.

There is no circularity concern. The cited theorems by the first author are published and independent, and the paper introduces no fitted constants. The example in the introduction showing sharpness is fine.

One minor thing: the text has a few typos in the proof (e.g. Wc vs WC), and the notation in the decomposition of W is a bit dense. Nothing that undermines the argument.

Who should read this: anyone working on regularity of powers of edge ideals. It settles a natural conjecture for bipartite graphs and gives the first universal bound for s=2. It deserves a serious referee; I'd send it out. The referee should mainly verify Theorem 3.1 line by line, and the rest will go through.

My recommendation: accept with minor revision, or at least send to peer review. If I were the editor, I would not desk reject.

Referee Report

2 major / 6 minor

Summary. This paper studies the Castelnuovo-Mumford regularity of powers of edge ideals. The main results are: (i) for any finite simple graph G, reg(I(G)^2) ≤ reg(I(G)) + 2; and (ii) for any bipartite graph G, reg(I(G)^s) ≤ 2s + reg(I(G)) − 2 for all s ≥ 2. The proof combines Hochster's formula with a topological inequality on clique complexes (Theorem 3.1) and an algebraic induction using colon ideals, polarization, and a theorem of Alilooee and Banerjee (Theorem 2.5). The bipartite bound is best possible, as complete bipartite graphs attain equality.

Significance. The bipartite statement (Theorem 1.1(ii)) is a substantial result: it proves a conjecture of Banerjee and others that the regularity of powers of bipartite edge ideals is bounded by 2s + reg(I(G)) − 2, and it shows the bound is tight. The proof introduces a new topological technique, suspension of clique complexes, to handle the base case s = 2. The paper is clearly organized and the algebraic induction for part (ii) is well structured. However, two gaps need to be addressed: a missing justification of the suspension identification in Theorem 3.1, and an erroneous algebraic inequality in the proof of part (i) for non-bipartite graphs.

major comments (2)

[Section 3, Theorem 3.1, Claim One] The identification H_l(Σ_{a,b}(Δ'[W_C])) = H_l(Δ[{a,b} ∪ W_C]) is asserted without proof. This equality is load-bearing: it is used to conclude that the third term in the first Mayer-Vietoris sequence vanishes, which is essential for the base case reg(I(G)^2 : ab) ≤ reg(I(G)). The equality is not immediate from the definitions; the authors should provide the argument that because W_C ⊆ C = A∩B, every vertex of W_C is non-adjacent in G to both a and b, so the added edges in G' do not affect W_C, and the induced complex Δ[{a,b} ∪ W_C] is exactly the suspension of Δ'[W_C]. The same reasoning is needed again in Claim Two.
[Section 3, proof of Theorem 1.1(i)] The step 'Let J' := I(G[V \ N(u)]), then (J : u) = J' + (variables) so reg(J : u) ≤ reg(J')' is false as stated. For example, take G to be the triangle with a leaf: vertices a,b,c,d and edges ab, ac, bc, cd. Then u = c lies in N(a)∩N(b), J = I(G), J' = I(G[{c}]) = 0, but (J : c) = (a,b,d) has regularity 1, so reg(J : u) ≤ reg(J') fails. This invalidates the proof of Theorem 1.1(i) for graphs where a common neighbor of an edge is adjacent to all other vertices. Since the bipartite case has N(a)∩N(b) = ∅, this error does not directly affect Theorem 1.1(ii), but the statement of (i) requires a corrected argument or a suitable restriction.

minor comments (6)

[Section 3, Theorem 3.1] The notation 'A ∪_C B' is not defined; it should be A ∪ B (or explained).
[Section 3, Theorem 3.1] The regularity of a simplicial complex reg(Δ) is used without definition; it should be defined as max{l+2 : H_l(Δ[W]) ≠ 0}, consistent with Theorem 2.7.
[Section 3, proof of Theorem 1.1(i)] In the short exact sequences, the index n is used where k should be, and the last sequence should involve u_k, not u_n.
[Throughout] There are several typos, including 'the varibles' in Section 2 and 'Froberg' in the introduction (should be 'Fröberg').
[References] Reference [26] is incomplete; it is listed as 'R. Woodrofe, J. Commut. Algebra 6, no. 2 (2014), 287–304' without a title.
[Section 4] In the discussion before Conjecture 4.3, the expression 'reg((I(G)^2 : ab) = 3' is missing a closing parenthesis.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the main inequality is proved by an in-paper topological bound plus prior published lemmas that do not already contain the target result.

full rationale

Walked the derivation chain. Theorem 1.1(i) is obtained by proving the colon bound reg(I(G)^2:ab) ≤ reg(I(G)) using polarization, short exact sequences, and the topological Theorem 3.1, which is established inside the paper from Hochster's formula and Mayer-Vietoris sequences. Theorem 1.1(ii) then follows by the colon recursive identity of Theorem 2.5 and Banerjee's induction Theorem 2.3; both are prior published results stated under hypotheses that do not include the target inequality. The recursive theorem only reduces reg(I(G)^s) to a colon bound plus 2s, and the paper supplies that missing bound. There are no fitted parameters, no normalization constants, and no quantity is 'predicted' from data that defines it. The one noteworthy point is Claim One's unproved identification H_l(Σ_{a,b}(Δ'[W_C])) = H_l(Δ[{a,b} ∪ W_C]); even if this is a gap in justification, it is a topological claim about the constructed complex and is not equivalent by definition to the main theorem. The self-citations to [1], [2], and [3] are published, independent, and do not contain Theorem 1.1, so they do not make the derivation circular.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on six standard theorems and a field assumption. No free parameters are fitted. No new entities are postulated. The main proof uses only established tools from combinatorial commutative algebra and algebraic topology.

assumptions (7)

standard math Hochster's formula (Theorem 2.7) expresses regularity of an edge ideal as the maximal l+2 over induced subcomplexes of the clique complex of the complement graph with nonzero reduced homology in degree l.
Used in Section 3 as the bridge between regularity and the topology of clique complexes.
standard math Froberg's theorem (Theorem 2.6(i)): the complement graph is chordal if and only if reg(I(G)) = 2.
Used to identify sharpness examples and in background discussion.
standard math Herzog-Hibi-Zheng (Theorem 2.6(ii)): for co-chordal graphs, reg(I(G)^s) = 2s for all s.
Provides the equality case for the sharpness claim.
standard math Banerjee's Theorem 2.3: reg(I(G)^{s+1}) is bounded by the maximum of reg(I(G)^s) and reg(I(G)^{s+1}:m_l)+2s over minimal generators m_l.
The algebraic engine that lets the proof reduce powers to colon ideals.
standard math Alilooee-Banerjee Theorem 2.5: for bipartite G, (I(G)^{s+1}:e_1...e_s) is an edge ideal of a bipartite graph on the same bipartition, and iterated colons have the stated form.
Essential for the induction step in part (ii); without it the reduction to an edge ideal of a bipartite graph fails.
standard math Woodrofe's lemma [26]: if two ideals are defined over disjoint variable sets, the regularity of their sum is the sum of their regularities minus one.
Used in the final estimate in part (i) to conclude reg(J') <= reg(I)-1.
domain assumption K is a field.
Standard assumption for the definition of Tor and regularity; no restrictive characteristics are assumed.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Regularity of Edge Ideals Via Suspension." pith.science (2026). https://pith.science/paper/OCVM5XOM

@misc{pith2026190803115,
  author       = {Pith},
  title        = {Pith review of: Regularity of Edge Ideals Via Suspension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OCVM5XOM}},
  note         = {Machine review of arXiv:1908.03115}
}

read the original abstract

We study the Castelnuovo-Mumford regularity of powers of edge ideals. We prove that if G is a bipartite graph, then reg(I(G)^s) \leq 2s + reg I(G) - 2 for all s \geq 2, which is the best possible upper bound for any s. Suspension plays a key role in proof of the base case s =2.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

Alilooee, A

A. Alilooee, A. Banerjee, Powers of Regularity Three of Bipartite Edge Ideals, J. Commut. Algebra 9 (2017), no. 4, 441–454

work page 2017
[2]

Banerjee, Regularity of Powers of Edge Ideals, J

A. Banerjee, Regularity of Powers of Edge Ideals, J. Algebraic C ombin. 41 (2015), no. 2, 303321

work page 2015
[3]

Regularity of powers of edge ideals: from local properties to global bounds

A. Banerjee, S. Bayerslan, H.T.Ha, Regularity of Powers of Edge Ideals: From Local Properties To Global Bounds, arxiv.org 1805.01434

work page Pith review arXiv
[4]

Banerjee, V.Mukundan, On The Powers of Unmixed Bipartite Ed ge Ideals, J.Alg and its Appln., To Appear

A. Banerjee, V.Mukundan, On The Powers of Unmixed Bipartite Ed ge Ideals, J.Alg and its Appln., To Appear

work page
[5]

Conca, Regularity jumps for powers of ideals

A. Conca, Regularity jumps for powers of ideals. Commutative Alg ebra with a focus on Geo- metric and Homological Aspects. Lect notes Pure and Appl., 244, Ch apman and Hall/CRC, Boca Raton, FL., 2006

work page 2006
[6]

Conca, J

A. Conca, J. Herzog, Castelnuovo-Mumford regularity of prod ucts of ideals, Collect. Math. 54 (2003), no. 2, 137–152

work page 2003
[7]

Cutkosky, J

S.D. Cutkosky, J. Herzog, and N.V. Trung, Asymptotic behaviou r of the Castelnuovo-Mumford regularity. Composito Mathematica 118 (1999), 243–261

work page 1999
[8]

Dao, C.Huneke J

H. Dao, C.Huneke J. Schweig, Bounds on the regularity and proje ctive dimension of ideals associated to graphs, J. Algebraic Combin. 38 (2013), 37–55

work page 2013

Show all 26 references

[9]

Dranishnikov , Boundaries of Coxeter groups and simplicial co mplexes with given links, J

A.N. Dranishnikov , Boundaries of Coxeter groups and simplicial co mplexes with given links, J. Pure and Appl. Alg., 137 (1999), 139–151

1999
[10]

Erey, Powers of ideals associated to ( C4,2K2)-free graphs, J

N. Erey, Powers of ideals associated to ( C4,2K2)-free graphs, J. Pure Appl. Algebra 223 (2019), no. 7, 3071-3080

2019
[11]

Erey, Powers of edge ideals with linear resolutions, Comm

N. Erey, Powers of edge ideals with linear resolutions, Comm. Alge bra 46 (2018), no.9, 4007- 4020

2018
[12]

Fr¨ oberg, On Stanley-Reisner rings, Topics in Algebra, Bana ch Center Publications, 26 (2) (1990), 57–70

R. Fr¨ oberg, On Stanley-Reisner rings, Topics in Algebra, Bana ch Center Publications, 26 (2) (1990), 57–70

1990
[13]

Tai Ha and A

H. Tai Ha and A. Van Tuyl, Resolution of square-free monomial id eals via facet ideals: a survey, Contemporary Mathematics 448 (2007), 91–117

2007
[14]

GTM 260, Springer–Verlag, 201 1

J.Herzog, T.Hibi: Monomial ideals. GTM 260, Springer–Verlag, 201 1
[15]

J.Herzog, T.Hibi, X.Zheng: Monomial ideals whose powers have a line ar resolution, Math. scand. 95 (2004), 23-32

2004
[16]

Jayanthan, N

A.V. Jayanthan, N. Narayanan and S. Selvaraja, Regularity of powers of bipartite graphs. J. Algebraic Combin. 47 (2018), no.1, 17-38

2018
[17]

S: Upper bounds for the regularity of powers of edge ideals of graphs, arXiv 1805.01412

A.V.Jayanthan, Selvaraja. S: Upper bounds for the regularity of powers of edge ideals of graphs, arXiv 1805.01412

arXiv
[18]

Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford r egularity

V. Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford r egularity. Proceedings of the American Mathematical Society (1999), 128 (2), 407–411

1999
[19]

Kummini, Homological invariants of monomial and binomial ideals, thesis, University of Kansas (2008)

M. Kummini, Homological invariants of monomial and binomial ideals, thesis, University of Kansas (2008)

2008
[20]

Miller and B

E. Miller and B. Sturmfels, Combinatorial commutative algebra, G raduate Texts in Mathemat- ics, Springer-Verlag, New York, 227 (2005)

2005
[21]

Morey and R

S. Morey and R. Villarreal, Edge ideals: algebraic and combinatoria l properties, Progress in Commutative Algebra: Ring Theory, Homology and Decomposition, de Gruyter, Berlin (2012), 85–126. REGULARITY OF EDGE IDEALS VIA SUSPENSION 9

2012
[22]

Nevo, Regularity of edge ideals of C4-free graphs via the topology of the lcm-lattice, J

E. Nevo, Regularity of edge ideals of C4-free graphs via the topology of the lcm-lattice, J. Combin. Theory Ser. A 118 (2011), 491–501

2011
[23]

Nevo and I

E. Nevo and I. Peeva, C4-free edge ideals J Algebraic Combin. 37 (2013), 243–248

2013
[24]

Przytycki and J

P. Przytycki and J. Swiatkowski, Flag-no-square triangulatio ns and Gromov boundaries in di- mension 3, Groups Geom. Dyn. 3 (2009), 453–468

2009
[25]

Raicu, Regularity and cohomology of determinantal thickenin gs, arXiv:1611.00415v2 (2017)

C. Raicu, Regularity and cohomology of determinantal thickenin gs, arXiv:1611.00415v2 (2017)

2017 arXiv
[26]

Woodrofe, J

R. Woodrofe, J. Commut. Algebra 6, no. 2 (2014), 287–304 Ramakrishna Mission Vivekananda Educational and Research Institute, Belur, India E-mail address : 123.arindam@gmail.com Einstein Institute of Mathematics, The Hebrew University o f Jerusalem. E-mail address : nevo@math.h...

2014

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alilooee, A

A. Alilooee, A. Banerjee, Powers of Regularity Three of Bipartite Edge Ideals, J. Commut. Algebra 9 (2017), no. 4, 441–454

work page 2017

[2] [2]

Banerjee, Regularity of Powers of Edge Ideals, J

A. Banerjee, Regularity of Powers of Edge Ideals, J. Algebraic C ombin. 41 (2015), no. 2, 303321

work page 2015

[3] [3]

Regularity of powers of edge ideals: from local properties to global bounds

A. Banerjee, S. Bayerslan, H.T.Ha, Regularity of Powers of Edge Ideals: From Local Properties To Global Bounds, arxiv.org 1805.01434

work page Pith review arXiv

[4] [4]

Banerjee, V.Mukundan, On The Powers of Unmixed Bipartite Ed ge Ideals, J.Alg and its Appln., To Appear

A. Banerjee, V.Mukundan, On The Powers of Unmixed Bipartite Ed ge Ideals, J.Alg and its Appln., To Appear

work page

[5] [5]

Conca, Regularity jumps for powers of ideals

A. Conca, Regularity jumps for powers of ideals. Commutative Alg ebra with a focus on Geo- metric and Homological Aspects. Lect notes Pure and Appl., 244, Ch apman and Hall/CRC, Boca Raton, FL., 2006

work page 2006

[6] [6]

Conca, J

A. Conca, J. Herzog, Castelnuovo-Mumford regularity of prod ucts of ideals, Collect. Math. 54 (2003), no. 2, 137–152

work page 2003

[7] [7]

Cutkosky, J

S.D. Cutkosky, J. Herzog, and N.V. Trung, Asymptotic behaviou r of the Castelnuovo-Mumford regularity. Composito Mathematica 118 (1999), 243–261

work page 1999

[8] [8]

Dao, C.Huneke J

H. Dao, C.Huneke J. Schweig, Bounds on the regularity and proje ctive dimension of ideals associated to graphs, J. Algebraic Combin. 38 (2013), 37–55

work page 2013