The edge ideal of a graph and its splitting graphs

J\"urgen Herzog; Masoomeh Rahimbeigi; Somayeh Moradi

The edge ideal of a graph and its splitting graphs

T0 review · 0 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 A splitting graph is a graph whose edges match a given graph while some vertices are duplicated, and this paper proves that for special splittings the edge ideal of the original graph has projective dimension and regularity no larger than…

desk verdict A genuinely new graph construction with a correct central inequality; the proof gaps are fillable exposition, not mathematical flaws. read the letter →

arxiv 1908.03769 v3 pith:NCTOO3AI submitted 2019-08-10 math.AC

Jürgen Herzog , Somayeh Moradi , Masoomeh Rahimbeigi This is my paper · ORCID

classification math.AC MSC 13F2013H10

keywords edgeidealssplittinggraphsprojectivedimensionCastelnuovo-MumfordregularityvertexidentificationBettinumbersmonomialgraphoperations

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

Given a graph $G$, a splitting graph $G'$ is a graph obtained by letting the edges of $G$ correspond bijectively to edges of $G'$ while some vertices of $G'$ map to the same vertex of $G$. The paper asks whether the edge ideal of $G$ is homologically simpler than the edge ideal of $G'$: specifically, whether $\operatorname{proj\,dim}(I(G)) \le \operatorname{proj\,dim}(I(G'))$ and $\operatorname{reg}(I(G)) \le \operatorname{reg}(I(G'))$. The paper proves these two inequalities for special splitting graphs, where identified vertices either have fully cross-adjacent neighborhoods or lie in different connected components. It also proves the inequalities for broad graph classes such as sequentially Cohen-Macaulay, chordal, weakly chordal, unmixed bipartite, very well-covered, and $C_5$-free vertex decomposable graphs. The upshot is that a purely combinatorial operation on graphs, also produced by the stretching operator from shifting theory, has controlled consequences for the homological invariants of edge ideals.

What carries the argument

The load-bearing object is the special splitting map, and the load-bearing identity is Lemma 1.4: for vertices $x,y$ with disjoint closed neighborhoods, $I(G):(x-y) = I(G) + (zw : z \in N_G(x), w \in N_G(y))$. Under special condition (1), the extra generators disappear because every neighbor of $x$ is adjacent to every neighbor of $y$, so the colon ideal equals $I(G)$. Substituting this equality into the short exact sequence yields both inequalities, since $x-y$ behaves as a nonzero divisor modulo the ideal of the graph obtained by identifying $x$ and $y$. Under special condition (2), the splitting graph splits into components and the same conclusion follows from the behavior of projective dimension and regularity for sums of ideals in disjoint variable sets.

What would settle it

Enumerate all special splitting graphs on at most eight vertices and directly compute $\operatorname{proj\,dim}$ and $\operatorname{reg}$ of the two edge ideals over the rationals; any pair violating $\operatorname{proj\,dim}(I(G)) \le \operatorname{proj\,dim}(I(G'))$ or $\operatorname{reg}(I(G)) \le \operatorname{reg}(I(G'))$ would falsify Theorem 1.3. A more surgical check is to perform the paper's iterative identifications and test at each step whether $I:(x-y)=I$; the first failure would pinpoint where the proof breaks.

Watch

Extended reading notes

Core claim

The paper's central discovery is Theorem 1.3: if $G'$ is a special splitting graph of $G$, then $\operatorname{proj\,dim}(I(G)) \le \operatorname{proj\,dim}(I(G'))$ and $\operatorname{reg}(I(G)) \le \operatorname{reg}(I(G'))$. A special splitting map is one in which any two vertices of $G'$ with the same image either have every neighbor of one adjacent to every neighbor of the other, or belong to different connected components of $G'$. The proof repeatedly identifies two such vertices and uses the short exact sequence $0 \to (S'/(I:x-y))(-1) \to S'/I \to S'/(I,x-y) \to 0$ to transfer the two inequalities from each identification step to the next. The paper further obtains the projective dimension inequality for sequentially Cohen-Macaulay graphs and the regularity inequality for several families in which regularity is governed by the matching number.

Load-bearing premise

The proof assumes that when two vertices lying over the same vertex are fused, and then the next such pair is fused, and so on, the two vertices being fused never acquire a common neighbor, so that the equality $I:(x-y)=I$ continues to hold at every step; this preservation is asserted rather than proved.

Editorial extensions

If this is right

For every special splitting graph, the original graph's edge ideal has projective dimension and regularity no larger than the splitting graph's, so fusing such duplicated vertices cannot create homological complexity.
If $G$ is sequentially Cohen-Macaulay, the projective dimension inequality holds for every splitting graph, because projective dimension equals the largest minimal vertex cover size and vertex covers lift to the splitting graph.
If $G$ is chordal, weakly chordal, sequentially Cohen-Macaulay bipartite, unmixed bipartite, very well-covered, or $C_5$-free vertex decomposable, the regularity inequality holds for every splitting graph, since in these families the regularity is $\nu(G)+1$ and induced matchings lift to the splitting graph.
For special splittings satisfying condition (2), the graded Betti number inequality $\beta_i(I(G)) \le \beta_i(I(G'))$ holds for all $i$.
For every graph and every splitting graph, $\dim(S'/I(G')) \ge \dim(S/I(G))$, and for paths and even cycles the same monotonicity holds for depth.

Reading between the lines

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

If the expected inequalities hold for all splittings rather than only special ones, then edge-preserving vertex identification of this kind would be a homological degeneration: projective dimension and regularity would be upper semicontinuous under contraction. A systematic computer search over all graphs with at most seven vertices could test this before any proof is attempted.
The paper's example showing that individual Betti numbers can decrease under splitting suggests that the true general statement, if one exists, concerns total Betti numbers or extremal invariants rather than componentwise comparison; comparing total Betti numbers across all small splittings is a concrete next experiment.
Because $\sigma$-stable graphs are splitting graphs produced by the stretching operator, determining whether those particular splittings are special, or proving the inequalities directly for them, would connect Theorem 1.3 to algebraic shifting and sharpen the picture of how stretching affects resolutions of edge ideals.

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

Dear colleague,

This paper introduces splitting graphs and proves that for special splittings, projective dimension and regularity are monotone: proj dim(I(G)) ≤ proj dim(I(G')) and reg(I(G)) ≤ reg(I(G')). That is a useful comparison tool, and the connection to Kalai's stretching operator gives it a natural home. The concept is new, and the main theorem is not a repackaging of known inequalities. Lemma 1.4 is carefully proved and is the right engine for the argument. Propositions 1.5–1.7 sensibly leverage known characterizations of graphs with extremal regularity or projective dimension, and the vertex-cover argument in 1.8 is clean. The paper is also honest that the general case for arbitrary splittings remains open, and the counterexample to a Betti-number inequality is helpful.

Soft spots are what you might expect from a paper that says “the others can be proved in the same way.” The induction in Theorem 1.3 for condition (1) asserts, rather than proves, that iterated vertex identifications preserve the special splitting property. I checked the one-step preservation; it works, with the delicate case being a shared neighbor of the two identified vertices. So the gap is fillable. The condition (2) argument silently changes polynomial rings; you need a sentence about extending scalars or passing to component rings, but the conclusion is right. Proposition 1.8's even-cycle case is dismissed as “similar” without detail—minor, but sloppy, especially since the path calculation itself has a few loose inequalities. There are small typos in Lemma 1.4's proof, but they do not affect the argument.

None of this undermines the main results. The paper deserves peer review, and a good referee should ask for the preserved-special lemma to be written out and for the even-cycle argument to be filled in. Both are doable.

Bottom line: a solid, citable contribution to edge ideal theory and shifting. I would bring it to a reading group and would cite it in work on graph invariants.

Best, [Your name]

Referee Report

0 major / 6 minor

Summary. The paper introduces the notion of a splitting graph of a given graph G, defined by a surjective vertex map whose induced edge map is bijective. The central result is Theorem 1.3, which states that if G' is a special splitting graph of G (satisfying one of two explicit conditions on fibers of the splitting map), then proj dim(I(G)) ≤ proj dim(I(G')) and reg(I(G)) ≤ reg(I(G')). The proof uses a colon-ideal lemma (Lemma 1.4) and a short exact sequence associated with identifying two vertices. The authors then give partial results for arbitrary splittings: regularity comparison when G is chordal, weakly chordal, sequentially Cohen-Macaulay bipartite, unmixed bipartite, very well-covered, or C5-free vertex decomposable (Proposition 1.6); projective dimension comparison when pd equals bight (Proposition 1.5) or when the splitting is special of type (2) (Proposition 1.7); dimension comparison for all graphs and depth comparison for paths and even cycles (Proposition 1.8); and closure of the bipartite and forest classes under arbitrary splittings (Propositions 1.10 and 1.11). The final section connects splitting graphs to the stretching operator from algebraic shifting theory.

Significance. If the result holds, the paper provides a new structural tool for comparing homological invariants of edge ideals under vertex identifications, a construction that appears naturally in algebraic shifting. The central theorem is substantive and the supporting lemmas are mostly self-contained; Lemma 1.4 is a clean and useful colon computation. The paper also gives explicit, falsifiable consequences for several well-studied graph classes, and the induced-matching lifting argument in Proposition 1.6 is elegant. The main claims are not obtained by curve fitting or circular reasoning; external citations are used only as genuine lower bounds. The main weaknesses are local exposition gaps rather than mathematical errors.

minor comments (6)

[Theorem 1.3, proof (i), condition (1)] The assertion that one can iterate the vertex-identification construction to obtain a sequence G' = G0, G1, ..., Gt = G with each Gi-1 a special splitting graph of Gi is not proved. This is a necessary induction step: after identifying x and y, one must verify that the new graph G1 still satisfies condition (1) of Definition 1.2 for all remaining pairs with equal image. In particular, the case where exactly one member of a remaining pair lies in NG0(y) requires a short argument. I believe the claim is true, but the proof should be supplied for completeness.
[Theorem 1.3, proof (i), condition (2)] The paragraph for condition (2) switches between the rings S and S' without making the identifications explicit. The ideal I(G) lives in S, not S', and the statement 'I(G'i) = I(Gi)' is an isomorphism after relabeling variables, not an equality. The projective dimension formulas should be written carefully, passing through the component rings and using the fact that projective dimension is invariant under polynomial extension when the variable sets are disjoint.
[Proposition 1.8] The sentence 'The argument for cycles of even length is similar' omits the whole proof for the even-cycle case. Since the depth inequality for even cycles is explicitly claimed, please either provide the argument or state the relevant projective dimension formulas for cycle graphs that allow the path-graph argument to be repeated.
[Theorem 1.3, proof (i), usage of Lemma 1.4] When applying Lemma 1.4 to conclude (I : x-y) = I, the paper does not state why the hypothesis NG0[x] ∩ NG0[y] = ∅ holds. It follows from condition (1) and the fact that the edge map is bijective, but this should be mentioned for clarity.
[Notation throughout] The paper alternates between proj dim(I(G)) and proj dim(S/I(G)) without comment. These quantities differ by 1 for nonzero proper ideals. Please make the notation consistent in the statements of Theorem 1.3, Propositions 1.5 and 1.7, and the introduction.
[Abstract and final section] There are minor typos: in the abstract, 'edge ideals of graph s' should read 'edge ideals of graphs', and in the discussion of the stretching operator, 'transforms u into a squarefree monomial ideal' should read 'squarefree monomial'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the central inequalities are proved from internal lemmas and independent external results, with no fitted parameters or self-referential reductions.

full rationale

The derivation chain of Theorem 1.3 uses internal constructions: the special splitting condition, Lemma 1.4 (proved in the paper), and the standard short exact sequence (3) with standard regularity and projective-dimension inequalities. The equality I:(x-y)=I is derived from Lemma 1.4 plus condition (1) of Definition 1.2, not assumed. The condition-(2) case invokes [3, Corollary 3.2], a published general bound on projective dimension of sums of ideals; although [3] is authored by one of the present authors, it is not the target result and is used only as an external inequality, so it is real evidence rather than a self-citation chain. Propositions 1.5 and 1.6 apply external characterizations (e.g., Morey-Villarreal, Katzman, Woodroofe) and prove the needed matching and vertex-cover comparisons inside the paper. No parameter is fitted and no prediction is renamed from an input. The only soft spot, the unexpanded induction that iterated vertex identifications preserve the special splitting property, is an exposition gap rather than a circular step; the preservation is verifiable directly and does not assume the conclusion. The paper is therefore self-contained against its own premises and receives score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper adds no free parameters and no new physical entities. It relies on standard polynomial ring theory, two cited lower bounds, and a small unproved graph-structural fact used in Proposition 1.8. The special splitting conditions are hypotheses of Theorem 1.3 rather than hidden axioms.

assumptions (5)

standard math Polynomial rings, minimal free resolutions, projective dimension, regularity, depth, and Betti numbers are standard.
Throughout the paper these objects are used with their standard definitions, not introduced here.
standard math bight(I(G)) - 1 <= pd(I(G)) (Morey-Villarreal [12, Cor 3.33]) and nu(G) + 1 <= reg(I(G)) (Katzman [8, Lemma 2.2]).
Proposition 1.5 and Proposition 1.6 use these as external lower bounds for arbitrary graphs.
standard math For an edge ideal, dim(S/I(G)) = |V(G)| - tau(G), where tau(G) is the minimum vertex cover size.
Used in Proposition 1.8 to prove the dimension inequality for every splitting graph.
standard math Sums of ideals satisfy pd(I+J) <= pd(I) + pd(J) + 1 and the Betti number convolution bound of [3].
Used in Theorem 1.3, condition (2), and Proposition 1.7 to compare projective dimensions and Betti numbers.
domain assumption Every splitting graph of a path graph is a disjoint union of path graphs; for even cycles the splitting graph is either the cycle or a forest of paths.
Stated without proof in Proposition 1.8; the depth computation for paths depends on this structural fact.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The edge ideal of a graph and its splitting graphs." pith.science (2026). https://pith.science/paper/NCTOO3AI

@misc{pith2026190803769,
  author       = {Pith},
  title        = {Pith review of: The edge ideal of a graph and its splitting graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NCTOO3AI}},
  note         = {Machine review of arXiv:1908.03769}
}

read the original abstract

We introduce and study the concept which we call the splitting of a graph and compare algebraic properties of the edge ideals of graphs and those of their splitting graphs.

Figures

Figures reproduced from arXiv: 1908.03769 by the authors.

**Figure 2.** An indecomposable splitting graph of G. We expect the following properties to hold. Let G′ be a splitting graph of G. Then (i) proj dim(I(G)) ≤ proj dim(I(G′ )); (ii) reg(I(G)) ≤ reg(I(G′ )); (iii) βi(I(G)) ≤ βi(I(G′ )) for all i. For the graded Betti numbers, an inequality as (iii) is not valid. Indeed, let G be the path graph with edges E(G) = {{1, 2}, {2, 3}} and G′ the splitting graph of G with edges E(G′ ) = {{… view at source ↗

**Figure 3.** A graph G and a splitting graph G′ of G. An independent set of G is a subset W ⊆ V (G) such that {i, j} * W for all edges {i, j} of G. Proposition 1.10. If G is a bipartite graph, then any splitting graph of G is so. Proof. Let G be a bipartite graph with the vertex partition X ∪ Y , where X and Y are independent sets of G. Consider any splitting graph G′ of G with the surjective map α: V (G′ ) → V (G) attached to i… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: G G′ b b b b b b b b b b [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Eliahou, M

S. Eliahou, M. Kervaire, Minimal resolutions of some monomial ideals , J. Algebra 129 (1990), 1–25

work page 1990
[2]

H. T. H` a, A. Van Tuyl, Monomial ideals, edge ideals of hypergraph s, and their graded Betti numbers, J. Algebraic Combin. 27, no. 2 (2008), 215–245

work page 2008
[3]

Herzog, A generalization of the Taylor complex construction, Communications in Algebra 35.5 (2007), 1747–1756

J. Herzog, A generalization of the Taylor complex construction, Communications in Algebra 35.5 (2007), 1747–1756

work page 2007
[4]

Herzog, T

J. Herzog, T. Hibi, Monomial ideals, Graduate Texts in Mathematic s. Springer, New York, (2010)

work page 2010
[5]

Jacques, Betti numbers of graph ideals, Thesis (2004)

S. Jacques, Betti numbers of graph ideals, Thesis (2004)

work page 2004
[6]

Kalai, Algebraic shifting, In: T

G. Kalai, Algebraic shifting, In: T. Hibi, (ed.) Computational commu tative algebra and com- binatorics, In: Adv. Studies in Pure Math. 33, Mathematical Society of Japan, Tokyo (2002)

work page 2002
[7]

Intersections of Leray complexes and regularity of monomial ideals

G. Kalai, R. Meshulam, Intersections of Leray complexes and reg ularity of monomial ideals, arXiv preprint math/0601745 (2006)

work page Pith review arXiv 2006
[8]

Katzman, Characteristic-independence of Betti numbers o f graph ideals, J

M. Katzman, Characteristic-independence of Betti numbers o f graph ideals, J. Combin. The- ory Ser. A 113 (2006), 435–454

work page 2006

Show all 14 references

[9]

Khosh-Ahang, S

F. Khosh-Ahang, S. Moradi, Regularity and projective dimension of the edge ideal of C5-free vertex decomposable graphs, Proceedings of the American Mathe matical Society 142 (2014), 1567–1576

2014
[10]

Kummini, Regularity, depth and arithmetic rank of bipartite ed ge ideals, Journal of Alge- braic Combinatorics 30, no

M. Kummini, Regularity, depth and arithmetic rank of bipartite ed ge ideals, Journal of Alge- braic Combinatorics 30, no. 4 (2009), 429

2009
[11]

Mahmoudi, A

M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai, and S. Yassemi, Vertex decom- posability and regularity of very well-covered graphs, Journal of P ure and Applied Algebra 215, no. 10 (2011), 2473–2480

2011
[12]

Morey, R

S. Morey, R. H. Villarreal, Edge ideals: Algebraic and combinatoria l properties, Progress in Commutative Algebra 1, (2012), 85-126

2012
[13]

Van Tuyl, Sequentially Cohen-Macaulay bipartite graphs: ver tex decomposability and regularity, Arch

A. Van Tuyl, Sequentially Cohen-Macaulay bipartite graphs: ver tex decomposability and regularity, Arch. Math. (Basel) 93, no. 5 (2009), 451–459

2009
[14]

Woodroofe, Matchings, coverings, and Castelnuovo-Mumf ord regularity, Journal of Com- mutative Algebra 6, no

R. Woodroofe, Matchings, coverings, and Castelnuovo-Mumf ord regularity, Journal of Com- mutative Algebra 6, no. 2 (2014), 287–304. J¨urgen Herzog, F achbereich Mathematik, Universit ¨at Duisburg-Essen, Campus Essen, 45117 Essen, Germany E-mail address : juergen.herzog@uni-ess...

2014

Pith tools

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

[1] [1]

Eliahou, M

S. Eliahou, M. Kervaire, Minimal resolutions of some monomial ideals , J. Algebra 129 (1990), 1–25

work page 1990

[2] [2]

H. T. H` a, A. Van Tuyl, Monomial ideals, edge ideals of hypergraph s, and their graded Betti numbers, J. Algebraic Combin. 27, no. 2 (2008), 215–245

work page 2008

[3] [3]

Herzog, A generalization of the Taylor complex construction, Communications in Algebra 35.5 (2007), 1747–1756

J. Herzog, A generalization of the Taylor complex construction, Communications in Algebra 35.5 (2007), 1747–1756

work page 2007

[4] [4]

Herzog, T

J. Herzog, T. Hibi, Monomial ideals, Graduate Texts in Mathematic s. Springer, New York, (2010)

work page 2010

[5] [5]

Jacques, Betti numbers of graph ideals, Thesis (2004)

S. Jacques, Betti numbers of graph ideals, Thesis (2004)

work page 2004

[6] [6]

Kalai, Algebraic shifting, In: T

G. Kalai, Algebraic shifting, In: T. Hibi, (ed.) Computational commu tative algebra and com- binatorics, In: Adv. Studies in Pure Math. 33, Mathematical Society of Japan, Tokyo (2002)

work page 2002

[7] [7]

Intersections of Leray complexes and regularity of monomial ideals

G. Kalai, R. Meshulam, Intersections of Leray complexes and reg ularity of monomial ideals, arXiv preprint math/0601745 (2006)

work page Pith review arXiv 2006

[8] [8]

Katzman, Characteristic-independence of Betti numbers o f graph ideals, J

M. Katzman, Characteristic-independence of Betti numbers o f graph ideals, J. Combin. The- ory Ser. A 113 (2006), 435–454

work page 2006