Bounds for the regularity of product of edge ideals

Arindam Banerjee; Priya Das; S. Selvaraja

REVIEW 2 major objections 4 minor 23 references

Bounds for the regularity of product of edge ideals

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Induced matchings bound the regularity of edge-ideal products

desk verdict New bounds for regularity of products of edge ideals, but the lower bound rests on an unproved Betti inequality that needs to be supplied before the exact formulas can be trusted. read the letter →

arxiv 1908.10573 v3 pith:K53FZ6FN submitted 2019-08-28 math.AC math.CO

Arindam Banerjee , Priya Das , S. Selvaraja This is my paper · ORCID

classification math.ACmath.CO MSC 13D0205E4505C70

keywords Castelnuovo-Mumfordregularityedgeidealsinducedmatchingnumberco-chordalcoverproductoflinearresolutionmonomial

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 the Castelnuovo-Mumford regularity of the product of two edge ideals, a numerical measure of how complicated the syzygies of an ideal are. For edge ideals I and J with I ⊆ J, coming from graphs H and G, it proves a lower bound in terms of the induced matching number common to both graphs, and an upper bound in terms of the co-chordal cover number of G and the regularity of I. When H sits inside G with equal induced matching numbers, the bounds collapse to an exact formula reg(IJ) = ν(G)+3 for several graph classes, including cycles on 3n vertices, weakly chordal graphs, and certain bipartite graphs. If the larger ideal J has a linear resolution, the paper shows the product has a linear resolution when reg(I) ≤ 4 and otherwise reg(IJ) = reg(I). These results turn a generally intractable invariant into a graph-counting problem for a natural class of monomial ideals.

What carries the argument

The load-bearing object is the colon ideal (IJ : ab), taken modulo a generator ab of I. Theorem 3.2 shows this colon is generated by quadrics, so after polarization it is again an edge ideal of a graph P that contains G. Lemma 3.4 proves co-chord(P) ≤ co-chord(G) by extending each co-chordal cover of G with carefully ordered edges, preserving the forbidden-induced-2K2 condition. This lets the proof feed the classical bound reg(I(G)) ≤ co-chord(G)+1 for a single edge ideal into the short exact sequences that relate IJ to its colon ideals. On the lower side, the comparison of graded Betti numbers between the square of the edge ideal of an induced matching and the full product IJ carries the argument, reducing the lower bound to ν_GH+3.

What would settle it

Compare the graded Betti numbers of I(Q)^2 and IJ for a pair of graphs H ⊆ G that share an induced matching Q using a computer algebra system; a single pair with β_{i,j}(I(Q)^2) > β_{i,j}(IJ) would invalidate the proof of the lower bound in Theorem 3.3.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: if I is the edge ideal of H and J is the edge ideal of G with I ⊆ J, then ν_GH + 3 ≤ reg(IJ) ≤ max{co-chord(G)+3, co-chord(H)+1}, where ν_GH is the induced matching number of H (equivalently of G when the matching is induced in both) and co-chord(−) is the minimum number of co-chordal subgraphs needed to cover the edge set. The paper also proves reg(IJ) ≤ max{reg(J)+3, reg(I)} as an incomparable upper bound. From these, it derives exact values for cycles with 3n vertices, weakly chordal graphs, unmixed bipartite graphs, and bipartite graphs of regularity 3, plus a dichotomy when J has a linear resolution.

Load-bearing premise

The lower bound requires that a known Betti-number comparison for powers of an edge ideal also holds for the product of two different edge ideals; the paper asserts this extension without proof, and if it fails the bound ν_GH+3 could break.

Editorial extensions

If this is right

For any subgraph H of G, reg(IJ) ≤ m(G)+3, where m(G) is the matching number of G (Corollary 3.6).
If H is an induced subgraph of G with ν(H)=ν(G), then reg(IJ)=ν(G)+3 whenever G is a cycle on 3n vertices, weakly chordal, unmixed bipartite, or bipartite with regularity 3 (Corollary 4.1).
If J has a linear resolution, then IJ has a linear resolution when reg(I) ≤ 4, and reg(IJ)=reg(I) when reg(I) ≥ 5 (Theorem 4.3).
For a chain of d=3 or 4 edge ideals with J_d the complete graph, the product has linear resolution if reg(J_1···J_{d−1}) ≤ 2d, and its regularity equals that of the shorter product otherwise (Theorem 4.6).
The upper bound reg(IJ) ≤ max{reg(J)+3, reg(I)} is sharp, as a 16-cycle with a perfect-matching subgraph shows (Example 3.10).

Reading between the lines

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

The Betti-number comparison used for the lower bound suggests a general principle: products of edge ideals may inherit lower bounds from the square of any induced subgraph ideal, so if the monotonicity fails, the lower bound may need a smaller constant or a modified combinatorial invariant.
The quadratic-generation result for colon ideals hints that a similar graph-cover description might work for products of more than two edge ideals without a complete-graph assumption, potentially yielding a recursive formula for reg(J_1···J_d).
The exact formulas for cycles and weakly chordal graphs likely extend to any graph where ν(G)=co-chord(G), so one could test whether reg(IJ)=ν(G)+3 holds for all gap-free graphs with H induced and ν equal.

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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

Two things to know. The paper proves genuinely new two-sided bounds for reg(IJ) when I and J are edge ideals with I⊆J, and works out exact formulas for several graph classes. The main theorem is: ν_GH + 3 ≤ reg(IJ) ≤ max{co-chord(G)+3, co-chord(H)+1}. That's a natural extension of the Katzman–Woodroofe bounds, and the upper bound argument is the real work: Theorem 3.2 identifies the colon ideal (IJ:ab) as a quadratic monomial ideal, and Lemma 3.4 shows the associated graph has co-chordal number no bigger than co-chord(G). That part is carefully done, with a long case analysis that looks correct to me. Theorem 3.9, relating reg(IJ) to reg(I) and reg(J), is also solid and gives a nice incomparable bound.

The soft spot is the lower bound. Theorem 3.3 imports Beyarslan–Hà–Trung's Lemma 4.2 — monotonicity of Betti numbers for powers under induced subgraphs — and asserts the same proof goes through for the product IJ. That is not automatic. Betti numbers are not monotone under ordinary inclusion of monomial ideals, so the mere inclusion P²⊆IJ is not enough. In the powers case there is an lcm-lattice injection; for products of two different ideals the pairs are (edge of H, edge of G) and the lcm of two edges of Q might also be realized by mixed pairs. The paper gives no argument. Since this lower bound feeds Corollaries 3.7, 4.1, 4.2, and Theorem 4.3(2), the exact formulas of Section 4 only hold if the Betti inequality actually extends. I think it likely does, but the authors need to prove it. Theorem 4.6 also ends with 'proceeding as in Theorem 4.3', which is thin for the d∈{3,4} claims, but that is secondary.

There are minor typos — Remark 4.2 writes ν(G)=ν(G) where ν(H)=ν(G) is meant — and the Macaulay2 examples are plausible checks.

Who's this for: people working on regularity of monomial ideals and edge ideals. It is a worthwhile contribution if the gap is filled. I'd send it to a serious referee, not desk reject, because the upper bound and the framework are real. The referee should ask for a complete proof of the lower bound as a revision condition.

Recommendation: engage with it, but condition your acceptance on fixing Theorem 3.3.

Referee Report

2 major / 4 minor

Summary. The paper studies the Castelnuovo-Mumford regularity of products of edge ideals IJ with I⊆J, where I and J are the edge ideals of graphs H and G. The main results are a lower bound in terms of a common induced matching number ν_GH and upper bounds in terms of co-chordal cover numbers and the regularities of I and J. These bounds are then applied to obtain exact formulas for several graph classes (cycles of length divisible by 3, weakly chordal graphs, unmixed bipartite graphs, graphs with dominating induced matchings) and to obtain linear-resolution criteria. A final theorem treats products of chains J1⊆⋯⊆Jd with Jd the edge ideal of a complete graph for d=3,4.

Significance. If the main lower bound were fully established, the paper would give a clean combinatorial description of reg(IJ), generalizing the Katzman-Woodroofe bounds for a single edge ideal and yielding exact formulas for natural graph classes. The upper-bound machinery via colon ideals and co-chordal covers (Lemma 3.4, Theorem 3.5) is original and carefully developed, and Theorem 3.9 is a useful inequality. The Macaulay2 examples are a concrete strength, and the statements are explicit and falsifiable. However, the central lower bound currently rests on an unproved Betti-number comparison, so the exact formulas of Section 4 are conditional on that missing argument.

major comments (2)

[Theorem 4.6 (Section 4)] The lower bound ν_GH+3≤reg(IJ) is not proved as written. The proof invokes Beyarslan-Hà-Trung Lemma 4.2, which compares Betti numbers of powers of an edge ideal when passing to an induced subgraph, and asserts that the proof 'goes through' for the product IJ. This is not a formality: the inclusion P^2⊆IJ does not by itself imply β_{i,j}(P^2)≤β_{i,j}(IJ), because Betti numbers are not monotone under inclusion of monomial ideals (e.g., reg(x^2,y^2)=3 but reg(x^2,xy,y^2)=2 even though (x^2,y^2)⊂(x^2,xy,y^2)). The lcm-lattice of P^2 embeds into that of IJ, but the desired inequality in the product case must be established explicitly, either by constructing a comparison of free resolutions or by proving the relevant lcm-lattice homology inequality. I note that Theorem 4.3(2) is proved by a separate exact-sequence argument and does not itself depend on this lower bound; however, the lower bound is load-bearing for Corollary 3.7, Corollary 4.1, Remark 4.2, and Theorem 4.3(1). This gap should be repaired before the paper is accepted.
[Theorem 4.6 (Section 4)] The proof of Theorem 4.6 is only a sketch and should be completed. The claim that every minimal generator of ((J,F1,…,F_{i-1}):F_i) has degree 2 is not derived in full detail from the preceding claim in all cases, and the final sentence 'Proceeding as in the proof of Theorem 4.3 we will get the desired conclusion' omits the actual Betti-number and regularity argument that is central to the theorem. Since the theorem asserts an exact value of regularity for products of three or four edge ideals, the long exact sequence and regularity comparison should be written out for d=3 and d=4, or the proof should be replaced by a reference to a complete argument.

minor comments (4)

[Section 2, definition of ν_GH] The definition of ν_GH as 'the largest size of induced matching of H as well as G' should be stated more explicitly, for example as the maximum size of a matching whose edges are edges of H and which is induced in G (and hence in H).
[Remark 4.2] In Remark 4.2, the condition 'ν(G)=ν(G)' should read 'ν(H)=ν(G)'; otherwise the statement is vacuous.
[Theorem 3.5 proof] In the proof of Theorem 3.5, the inequality 'reg( ~IJ : fi) ≤ co-chord(P_i)+1' is stated for ideal regularity, but the exact-sequence chain above it requires a bound on the quotient regularity reg(R/(IJ:f_i)). Using the quotient version reg(R/I(P_i)) ≤ co-chord(P_i) gives the stated upper bound; as written this appears to be an off-by-one slip in the presentation, not in the result, but it should be corrected.
[Abstract and Introduction] There are several typographical errors ('Caste lnuovo-Mumford', 'regular ity', 'SEL V ARAJA') that should be corrected in a final version.

Circularity Check

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No circularity: the regularity bounds follow from independent external theorems and standard exact-sequence arguments; the suspected Betti-monotonicity gap is a correctness issue, not a circular reduction.

full rationale

The derivation chain is not circular. The paper obtains its lower and upper bounds (Theorems 3.3 and 3.5) from standard short exact sequences, the colon-ideal structure theorem (Theorem 3.2), and external results: Woodroofe's bound reg(I(G)) <= co-chord(G)+1, Beyarslan-Ha-Trung's formula for regularity of powers, Froberg's linear-resolution theorem, and Jacques's induced-subgraph regularity bound. Theorem 3.3's use of [4, Lemma 4.2] is not a definitional reduction: the paper asserts, rather than proves, that the Betti monotonicity for powers extends to products IJ, and that unproved transfer is a correctness gap if it fails, but it is not an equation that makes the conclusion equal to its input. Corollary 4.1 uses [17] (same-author work) for nu(G)=co-chord(G) in certain graph classes; that cited result is an independent theorem with stated hypotheses, not a restatement of the target regularity formula. There is no fitted parameter later renamed a prediction, no 'uniqueness theorem' imported from the authors to rule out alternatives, and no ansatz smuggled in by self-citation. The reviewer-identified Betti-monotonicity gap affects soundness, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted. The graph invariants such as nu, co-chord, min-match, and m are combinatorial quantities, not adjustable constants. The paper introduces no new algebraic or geometric objects; its new content consists of inequalities and exact formulas built from standard edge ideals and their products.

assumptions (6)

standard math Katzman-Woodroofe bound: nu(G)+1 <= reg(I(G)) <= co-chord(G)+1
Used as the upper bound for reg(I) in Theorem 3.5 and elsewhere; cited from reference [23].
standard math Beyarslan-Ha-Trung Lemma 4.2: Betti numbers of powers of an induced subgraph ideal are bounded by those of the larger graph
The lower bound Theorem 3.3 depends on extending this comparison to the product IJ; cited from reference [4].
standard math Froberg's theorem: an edge ideal has a linear resolution if and only if its graph is co-chordal
Used in Theorem 4.3 and Theorem 4.6; cited from reference [12].
standard math Polarization preserves regularity of monomial ideals
Used repeatedly to pass from (IJ:ab) to an associated graph; cited from Herzog and Hibi, Corollary 1.6.3(a).
standard math Standard short exact sequence and regularity inequality for colon ideals
The proofs of Theorems 3.5, 3.9, 4.3 and 4.6 all use the exact sequences displayed in equation (3.2) and the inequality from reference [18], Theorem 1.2.
standard math Known classification results bounding reg(I) <= 4 for graph classes such as gap-cricket-free, gap-diamond-free, and gap-C4-free graphs
Used in Corollaries 3.11 and 4.5; cited from references [1], [10], [11], and [20].

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Pith. "Pith review of Bounds for the regularity of product of edge ideals." pith.science (2026). https://pith.science/paper/K53FZ6FN

@misc{pith2026190810573,
  author       = {Pith},
  title        = {Pith review of: Bounds for the regularity of product of edge ideals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K53FZ6FN}},
  note         = {Machine review of arXiv:1908.10573}
}

abstract

Let $I$ and $J$ be edge ideals in a polynomial ring $R = \mathbb{K}[x_1,\ldots,x_n]$ with $I \subseteq J$. In this paper, we obtain a general upper and lower bound for the Castelnuovo-Mumford regularity of $IJ$ in terms of certain invariants associated with $I$ and $J$. Using these results, we explicitly compute the regularity of $IJ$ for several classes of edge ideals. Let $J_1,\ldots,J_d$ be edge ideals in a polynomial ring $R$ with $J_1 \subseteq \cdots \subseteq J_d$. Finally, we compute the precise expression for the regularity of $J_1 J_2\cdots J_d$ when $d \in \{3,4\}$ and $J_d$ is the edge ideal of complete graph.

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

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

[1] [1]

Banerjee

A. Banerjee. The regularity of powers of edge ideals. J. Algebraic Combin. , 41(2):303–321, 2015

work page 2015

[2] [2]

Banerjee, S

A. Banerjee, S. K. Beyarslan, and H. Huy T` ai. Regularity of edge ideals and their powers. In Advances in algebra , volume 277 of Springer Proc. Math. Stat. , pages 17–52. Springer, Cham, 2019

work page 2019

[3] [3]

Benzaken, Y

C. Benzaken, Y. Crama, P. Duchet, P. L. Hammer, and F. Maﬀr ay. More characterizations of triangulated graphs. J. Graph Theory, 14(4):413–422, 1990

work page 1990

[4] [4]

Beyarslan, H

S. Beyarslan, H. T. H` a, and T. N. Trung. Regularity of pow ers of forests and cycles. J. Algebraic Combin. , 42(4):1077– 1095, 2015

work page 2015

[5] [5]

Caviglia

G. Caviglia. Bounds on the Castelnuovo-Mumford regular ity of tensor products. Proc. Amer. Math. Soc. , 135(7):1949– 1957, 2007

work page 1949

[6] [6]

Chardin, N

M. Chardin, N. C. Minh, and N. V. Trung. On the regularity o f products and intersections of complete intersections. Proc. Amer. Math. Soc. , 135(6):1597–1606, 2007

work page 2007

[7] [7]

Cimpoea¸ s

M. Cimpoea¸ s. Some remarks on Borel type ideals. Comm. Algebra , 37(2):724–727, 2009

work page 2009

[8] [8]

Conca and J

A. Conca and J. Herzog. Castelnuovo-Mumford regularity of products of ideals. Collect. Math. , 54(2):137–152, 2003

work page 2003