Joins, Ears and Castelnuovo-Mumford regularity

Jorge Neves; Maria Vaz Pinto; Rafael H. Villarreal

REVIEW

Joins, Ears and Castelnuovo-Mumford regularity

Jorge Neves, Maria Vaz Pinto, Rafael H. Villarreal

Commutative Algebra · Submitted 6 Sep 2019 · 35 references

Pith's verdict: A new graph ideal with a clean bipartite regularity formula, but the advertised ear-decomposition corollary is hostage to an unpublished companion paper.

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arXiv ID1909.02773 (version 1)

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CitationJorge Neves, Maria Vaz Pinto, Rafael H. Villarreal. "Joins, Ears and Castelnuovo-Mumford regularity." arXiv:1909.02773 (2019). https://pith.science/paper/Q2IING2F

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Editor's note

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I'll get straight to it. The main result, reg I(X_G) = mu(G) for bipartite G, is real and the proof is self-contained. The construction of I(X_G) as the preimage of (x_i^2 - x_j^2) under t_e maps to x_i x_j is new, and the paper carefully develops its basics: one-dimensional Cohen-Macaulay, binomial, field-independent initial ideal, and coincidence with the vanishing ideal over Z_3. The lower bound mu(G) <= reg I(X_G) has a direct graph-theoretic proof using joins and Proposition 2.5, and the upper bound is standard but cleanly transferred. Proposition 3.2 gives a useful package of regularity results for special graphs, and the bipartite equality proof is the real contribution; it handles the case analysis with even cycles and blocks well. I believe the math is correct, modulo small errors.

The soft spots are fixable but real. The advertised new combinatorial result, Corollary 4.12, is not self-contained: Proposition 4.11 invokes [24, Theorem 4.4], a to-appear paper by the first author, and no statement is reproduced. If that theorem is unavailable or wrong, the corollary falls, although the main equality does not depend on it. This should be fixed by adding a proof or a precise statement of the theorem. There are also two sign typos: in the proof of Proposition 4.3 the displayed equality has |V_H| - b_0(H) - 1 where it should be +1 (Proposition 3.2(vi) is stated correctly), and in the proof of Proposition 2.5 the phrase 'x^delta + x^gamma' should read 'x^delta - x^gamma'. Both are harmless to the final claims, but they will confuse a reader.

On citation practice: the paper leans heavily on the vanishing ideal literature, including several papers by the same authors. That is not a flaw here because the transfers are explicit and the cited results are the right tools; the only problematic citation is the to-appear one.

Who is this for? Commutative algebraists and algebraic combinatorialists working on graph ideals, vanishing ideals, and regularity. It deserves a serious referee: the main theorem is new and the proof is rigorous. With the typos fixed and the [24] dependency addressed, I would happily accept. My advice: send it out, ask the authors to state [24, Theorem 4.4] or prove the needed case, and let a knowledgeable referee check the bipartite proof.

Referee report

Significance

If the results hold, the paper provides a new class of graph ideals with a clean algebraic invariant: the regularity equals the maximum vertex join number for bipartite graphs. The proof of the lower bound in Theorem 4.2 is a direct graph-theoretic argument with no fitted parameters, and the proof of Theorem 4.5 is self-contained and does not rely on the to-appear paper. The structural results (Cohen-Macaulayness, binomiality, field independence) are useful and transfer vanishing-ideal regularity results to this new setting. The advertised combinatorial consequence about nested ear decompositions is interesting, but it is not established within the manuscript because it depends entirely on [24, Theorem 4.4], which is not reproduced. Thus the significance of the central algebraic theorem is solid, while the significance of the stated ear-decomposition corollary is currently conditional.

Major objections (2)

[4.2, Proposition 4.11 and Corollary 4.12] The proof of Proposition 4.11 imports [24, Theorem 4.4], a to-appear paper by the first author, and uses it with K = Z_3 to assert that reg I(X_G) = (|V_G| + epsilon - 1)/2 for a bipartite nested ear decomposition. No statement or proof of [24, Theorem 4.4] is included in the preprint, so the derivation of Corollary 4.12, which is advertised in the abstract as a new combinatorial result, is not verifiable from this manuscript alone. This is load-bearing for that corollary, although it does not affect Theorem 4.5, whose proof is self-contained. Please include the statement of [24, Theorem 4.4] (or a proof), or explicitly mark Corollary 4.12 as conditional on that result.
[4.1, Proposition 4.3, proof] The final displayed equality in the proof of Proposition 4.3 reads reg I(X_H) = |V_H| - b_0(H) - 1 for the non-bipartite graph H with a single odd cycle; the correct value is |V_H| - b_0(H) + 1, as stated in (9) and in Proposition 3.2(vi). As written, the proof concludes the false bound reg I(X_G) <= |V_G| - b_0(G) - 1. This is an easily fixable sign typo, but it should be corrected because it appears in the proof of a central bound.

Minor objections (4)

[2.4, Proposition 2.5, proof] In the proof of the claim, the text says 'let us show that x^delta + x^gamma belongs to the ideal'; the binomial under consideration is x^delta - x^gamma, and the same sign typo appears again a few lines below. The argument is correct once the signs are fixed.
[3, Proposition 3.5, statement] The statement begins 'Let t^delta be a monomial. Then, t^alpha in (I(X_G), t^delta) if and only if...' but the monomial t^alpha is not introduced. It should read 'Let t^delta and t^alpha be monomials in K[E_G]'.
[4.1, Theorem 4.5, proof] After deriving inequality (11), the proof says 'Without loss of generality, we may then assume that C is an even cycle.' This reduction is valid because averaging over the edge-disjoint even cycles produced by the bipartite circuit forces one constituent cycle to satisfy the strict inequality, but this averaging step should be stated explicitly.
[General] The manuscript contains several typographical artifacts, such as 'W e' in the abstract and 'deﬁned' with a ligature, and would benefit from a careful proofreading pass.

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The paper

The paper claims

For bipartite graphs, the Castelnuovo–Mumford regularity of the graph ideal $I(X_G)$ equals the maximum vertex join number, and this equality forces a new invariant property of nested ear decompositions.

The reading

This paper introduces a new binomial ideal attached to any simple graph: the ring map $t_e\mapsto x_ix_j$ pulls the ideal generated by the differences $x_i^2-x_j^2$ back from the vertex ring to the edge ring. The quotient is a one-dimensional Cohen–Macaulay graded ring, and the ideal's initial ideal can be generated independently of the field. The main claim is that its Castelnuovo–Mumford regularity is bounded by $\mu(G)\le \operatorname{reg} I(X_G)\le |V_G|-b_0(G)+1$, and that for bipartite $G$ the lower bound is exact: $\operatorname{reg} I(X_G)=\mu(G)$. This matters because it converts a generally hard algebraic invariant into a purely combinatorial optimization problem over circuits, and because the equality yields a combinatorial consequence: every nested ear decomposition of a bipartite graph has exactly the minimum possible number of even-length ears.

What carries the argument

The load-bearing object is the binomial ideal $I(X_G)$, defined as the preimage of $(x_i^2-x_j^2: i,j\in V_G)$ under the monomial map sending each edge variable $t_{\{i,j\}}$ to $x_ix_j$. Two structural facts carry the argument: over $\mathbb{Z}_3$ this ideal coincides with the vanishing ideal of the projective toric subset parameterized by $G$, and with respect to a fixed monomial order its initial ideal has generators independent of the field, so regularity can be transferred from known finite-field results. On the combinatorial side, the proof uses the maximum vertex join number $\mu(G)$ together with a monomial membership criterion (Proposition 3.5) to show that one-edge Artinian reductions vanish exactly when the degree reaches $\mu(G)$; inside that argument, a subgraph in which every vertex has even degree decomposes into edge-disjoint cycles.

Extended reading notes

Core claim

The central discovery is Theorem 4.5: for every bipartite graph $G$ without isolated vertices, the Castelnuovo–Mumford regularity of $I(X_G)=\theta^{-1}(x_i^2-x_j^2: i,j\in V_G)$ equals the maximum vertex join number $\mu(G)$, the largest set of edges that contains at most half of the edges of every circuit. The proof also establishes companion bounds valid for every simple graph, $\mu(G)\le \operatorname{reg} I(X_G)\le |V_G|-b_0(G)+1$, with the upper bound sharpened to $|V_G|-b_0(G)$ in the bipartite case. Using a cited result on vanishing ideals over nested ear decompositions—ear decompositions in which each new path is nested in an earlier path, with nest intervals that are disjoint or one contained in the other—the paper derives that in a bipartite graph every nested ear decomposition has the same number of even-length ears, namely $\phi(G)$, the minimum over all ear decompositions of the graph.

Load-bearing premise

The load-bearing premise is that the cited to-appear theorem [24, Theorem 4.4] correctly computes the regularity of the vanishing ideal over a bipartite graph with a nested ear decomposition; the ear-counting corollary depends on it, even though the main equality for bipartite graphs has a self-contained proof.

Editorial extensions

If this is right

For bipartite graphs, computing $\operatorname{reg} I(X_G)$ becomes a finite combinatorial problem: find the largest edge set $J$ with $|J\cap E_C|\le |E_C|/2$ for every circuit $C$.
The regularity bounds are sharp: forests attain $|V_G|-b_0(G)$, graphs with a single odd cycle attain $|V_G|-b_0(G)+1$, complete bipartite graphs $K_{a,b}$ have regularity $\max\{a,b\}$, and Hamiltonian bipartite graphs have regularity $|V_G|/2$.
In a bipartite graph with a nested ear decomposition, the number of even-length ears is the same in every such decomposition and equals the global minimum $\phi(G)$ over all ear decompositions.
Since $\operatorname{reg} I(X_G)$ is independent of the field, the equality $\operatorname{reg} I(X_G)=\mu(G)$ holds over every field, with finite-field vanishing ideals serving only as a transfer device.

Reading between the lines

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

The same monomial-membership technique that proves the bipartite equality might be tested on non-bipartite graphs with controlled odd-cycle structure; the gap between the upper and lower bounds in the general case suggests that some statistic finer than $\mu(G)$ would be needed there.
Because $I(X_G)$ is field-independent and agrees with the vanishing ideal over $\mathbb{Z}_3$, the equality gives a characteristic-free way to compute a regularity value that appears in the theory of parameterized linear codes associated to graphs.
A natural testable extension is to ask whether the ear-counting corollary persists for wider classes of ear decompositions, or whether nested is essential; the paper's proof of invariance uses the cited nested-case regularity formula, not a direct ear-by-ear argument.

Original abstract

We introduce a new class of polynomial ideals associated to a simple graph, $G$. Let $K[E_G]$ be the polynomial ring on the edges of $G$ and $K[V_G]$ the polynomial ring on the vertices of $G$. We associate to $G$ an ideal, $I(X_G)$, defined as the preimage of $(x_i^2-x_j^2 : i,j\in V_G)\subseteq K[V_G]$ by the map $K[E_G]\to K[V_G]$ which sends a variable, $t_e$, associated to an edge $e=\{i,j\}$, to the product $x_ix_j$ of the variables associated to its vertices. We show that $K[E_G]/I(X_G)$ is a one-dimensional, Cohen-Macaulay, graded ring, that $I(X_G)$ is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field $K$. We focus on the Castelnuovo-Mumford regularity of $I(X_G)$ providing the following sharp upper and lower bounds: $$ \mu(G) \leq \operatorname{reg} I(X_G) \leq |V_G|-b_0(G)+1, $$ where $\mu(G)$ is the maximum vertex join number of the graph and $b_0(G)$ is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We introduce a new class of polynomial ideals associated to a simple graph, $G$. Let $K[E_G]$ be the polynomial ring on the edges of $G$ and $K[V_G]$ the polynomial ring on the vertices of $G$. We associate to $G$ an ideal, $I(X_G)$, defined as the preimage of $(x_i^2-x_j^2 : i,j\in V_G)\subseteq K[V_G]$ by the map $K[E_G]\to K[V_G]$ which sends a variable, $t_e$, associated to an edge $e=\{i,j\}$, to the product $x_ix_j$ of the variables associated to its vertices. We show that $K[E_G]/I(X_G)$ is a one-dimensional, Cohen-Macaulay, graded ring, that $I(X_G)$ is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field $K$. We focus on the Castelnuovo-Mumford regularity of $I(X_G)$ providing the following sharp upper and lower bounds: $$ \mu(G) \leq \operatorname{reg} I(X_G) \leq |V_G|-b_0(G)+1, $$ where $\mu(G)$ is the maximum vertex join number of the graph and $b_0(G)$ is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We introduce a new class of polynomial ideals associated to a simple graph, $G$.	contribution 1.00	0.95	0.04	asserted 0.98
s1	Let $K[E_G]$ be the polynomial ring on the edges of $G$ and $K[V_G]$ the polynomial ring on the vertices of $G$.	background 0.84	0.17	0.02	asserted 0.85
s2	We associate to $G$ an ideal, $I(X_G)$, defined as the preimage of $(x_i^2-x_j^2 : i,j\in V_G)\subseteq K[V_G]$ by the map $K[E_G]\to K[V_G]$ which sends a variable, $t_e$, associated to an edge $e=\{i,j\}$, to the product $x_ix_j$ of the variables associated to its vertices.	contribution 0.67	0.88	0.04	asserted 0.98
s3	We show that $K[E_G]/I(X_G)$ is a one-dimensional, Cohen-Macaulay, graded ring, that $I(X_G)$ is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field $K$.	result 0.99	0.96	0.19	asserted 1.00
s4	We focus on the Castelnuovo-Mumford regularity of $I(X_G)$ providing the following sharp upper and lower bounds: $$ \mu(G) \leq \operatorname{reg} I(X_G) \leq \|V_G\|-b_0(G)+1, $$ where $\mu(G)$ is the maximum vertex join number of the graph and $b_0(G)$ is the number of its connected components.	result 0.96	0.96	0.65	asserted 1.00
s5	We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.	result 0.88	0.96	0.23	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
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levels run

P abstract

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T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** A non-bipartite graph. nomials together with t 2 ij − t 2 46 : {i,j} ∈ EG \ {{4,6}}

**Figure 2.** A Hamiltonian bipartite graph. P1 of an ear decomposition starting from P0 = 1. The remaining ears are all edges, P2 = {2, 3} and P3 = {1, 4}, for example, in this order. Another ear decomposition of G is given by: P0 = 1, P1 = (1, 2, 3, 4, 1), P2 = (2, 5, 4), P3 = (3, 6, 1). Whereas the first ear decomposition has a single even length ear, the second has three

Figures reproduced from arXiv:1909.02773 by the authors.

References

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [24]

J. Neves, Regularity of the vanishing ideal over a bipartite nested ea r decomposition, J. Al- gebra Appl., 28 pp. (to appear)

View on Pith

[2] [1]

J. Biermann, A. O’Keefe and A. Van Tuyl, Bounds on the regularity of toric ideals of graphs , Adv. Appl. Math. 85 (2017), 84–102

View on Pith 2017

[3] [2]

S. Beyarslan, H. T. H` a and T. N. Trung, Regularity of powers of forests and cycles , J. Alge- braic Combin. 42 (2015), no. 4, 1077–1095. JOINS, EARS AND CASTELNUOVO–MUMFORD REGULARITY 17

View on Pith 2015

[4] [3]

B. Bollob´ as, Modern graph theory , Graduate Texts in Mathematics, 184, Springer-Verlag, New York, 1998

View on Pith 1998

[5] [4]

D. Eppstein, Parallel Recognition of Series-Parallel Graphs , Information and Computation, Volume 98, Issue 1, (1992), 41–55

View on Pith 1992

[6] [5]

D. Eisenbud, The geometry of syzygies. A second course in commutative alg ebra and algebraic geometry, Graduate Texts in Mathematics, 229, Springer-Verlag, New York, 2005

View on Pith 2005

[7] [6]

V. Ene and A. Zarojanu, On the regularity of binomial edge ideals , Math. Nachr. 288 (2015), no. 1, 19–24

View on Pith 2015

[8] [7]

A. Frank, Conservative weightings and ear-decompositions of graphs , Combinatorica 13 (1993), no. 1, 65–81

View on Pith 1993

[9] [8]

M. Gonz´ alez and C. Renter ´ ıa, Evaluation codes associated to complete bipartite graphs , Int. J. Algebra, 2 (2008), no. 1–4, 163–170

View on Pith 2008

[10] [9]

M. Gonz´ alez, C. Renter ´ ıa and E. Sarmiento,Parameterized codes over some embedded sets and their applications to complete graphs , Math. Commun., 18 (2013), no. 2, 377–391

View on Pith 2013

[11] [10]

H. T. H` a, S. K. Beyarslan and A. O’Keefe, Algebraic properties of toric rings of graphs , Comm. Algebra, 47 (2019), no. 1, 1–16

View on Pith 2019

[12] [11]

H. T. H` a and A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their grad ed Betti numbers , J. Algebraic Combin. 27 (2008), no. 2, 215–24

View on Pith 2008

[13] [12]

T. Hibi, A. Higashitani, K. Kimura and A. O’Keefe, Algebraic study on Cameron-Walker graphs, J. Algebra 422 (2015), 257–269

View on Pith 2015

[14] [13]

T. Hibi, A. Higashitani, K. Kimura and A. Tsuchiya, Dominating induced matchings of ﬁnite graphs and regularity of edge ideals , J. Algebraic Combin. 43 (2016), no. 1, 173–198

View on Pith 2016

[15] [14]

H. Higashidaira, On the sequentially Cohen-Macaulay properties of almost co mplete multi- partite graphs., Comm. Algebra 45 (2017), no. 6, 2478–2493

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