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arxiv: 2607.00838 · v1 · pith:PRRPGOCNnew · submitted 2026-07-01 · 🧮 math.AC · math.CO

On the Linearity of Squarefree Powers of Edge Ideals

Francesco Navarra , Ayesha Asloob Qureshi , Naoki Terai This is my paper

Pith reviewed 2026-07-02 01:48 UTC · model grok-4.3

classification 🧮 math.AC math.CO
keywords edge idealssquarefree powerslinearly relatedBetti tableslinear resolutionsmatchingsflag simplicial complexesStanley-Reisner ideals
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The pith

Matchings of size p characterize when the squarefree power of an edge ideal is linearly related.

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

The paper establishes a combinatorial criterion, based on matchings in the graph, that determines whether the first syzygy module of I(G)^{[p]} is generated by linear forms. This reduces questions about the algebraic structure of the ideal to properties of the underlying graph. For the Stanley-Reisner ideal of a 1-dimensional flag simplicial complex, the work further specifies the exact shape of the Betti table and identifies when the resolution is linear.

Core claim

The p-th squarefree power I(G)^{[p]} is the monomial ideal generated by squarefree monomials corresponding to the matchings of size p of G. We provide a combinatorial characterization of when I(G)^{[p]} is linearly related. For a 1-dimensional flag simplicial complex Δ and its Stanley-Reisner ideal I_Δ, which arises as the edge ideal of the complement graph of Δ, we describe the shape of the Betti table of I_Δ^{[p]} and give a combinatorial characterization of when I_Δ^{[p]} has a linear resolution.

What carries the argument

The p-sized matchings of G, which generate I(G)^{[p]} and translate the algebraic condition of linear relatedness into graph-theoretic terms.

If this is right

  • When the matching condition holds, the first syzygy module of I(G)^{[p]} is generated by linear forms.
  • For 1-dimensional flag complexes the Betti table of I_Δ^{[p]} takes a specific explicit shape determined by the p-matchings.
  • I_Δ^{[p]} has a linear resolution precisely when an additional combinatorial condition on the complex holds.

Where Pith is reading between the lines

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

  • The matching-based criterion may let researchers decide homological properties of these ideals by direct graph inspection rather than Gröbner-basis calculations.
  • The same translation technique could be tested on squarefree powers of other monomial ideals arising from simplicial complexes.
  • If the characterization extends, it would link the existence of induced matchings directly to the projective dimension of the ideal.

Load-bearing premise

The combinatorial conditions on matchings of size p control the syzygies without additional hidden algebraic relations interfering.

What would settle it

A graph G satisfying the stated matching condition for which the first syzygy module of I(G)^{[p]} still requires generators of degree greater than one.

Figures

Figures reproduced from arXiv: 2607.00838 by Ayesha Asloob Qureshi, Francesco Navarra, Naoki Terai.

Figure 1
Figure 1. Figure 1: A complete bipartite K2,4 and a crown graph Cr(5). Let ∆ be a simplicial complex on [n], and let R = K[x1, . . . , xn] be the polynomial ring in n variables over a field K, where deg(xi) = 1 for all i ∈ [n]. Throughout this paper, K denotes an arbitrary field unless otherwise specified. A subset F ⊆ [n] is called a non-face of ∆ if F /∈ ∆. We denote by N (∆) the collection of minimal non-faces of ∆. For an… view at source ↗
Figure 2
Figure 2. Figure 2: A 1-dimensional simplicial complex and its complement graph [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Graphs in the Example 3.2. If ∆ is a 1-dimensional flag simplicial complex, since I∆ = I(∆), we may equivalently regard I∆ as the edge ideal of a graph. Thus, the property of being linearly related I [p] ∆ , stated below, follows as a special case of Theorem 3.1 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A 1-dimensional flag simplicial complex. By Corollary 3.3, I [2] ∆ is not linearly related, since, for instance, the induced subgraph of ∆ on the vertex set {1, 2, 4, 5, 6, 7} is isomorphic to K2,4. Using Macaulay2 [GS], the Betti table of I [2] ∆ [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example for p = 5 and i = 7. We can now compute δ2p−1(Γ). Observe first that, in the expression of δ2p−1(Γ), the sum of the terms obtained by deleting two primed vertices is zero because it is the result of the application of two simplicial differentials. Therefore, it suffices to consider the terms obtained by deleting exactly one non-primed vertex and one primed vertex, that is, in other words, the 3 × (… view at source ↗
Figure 6
Figure 6. Figure 6: Example for p = 5 and i = 7: terms in δ2p−1(Γ). The occurrence given by γ1 can occur only in an another alternative way in δ2p−1(Γ), that we now construct. Set He = (H \ {hr})∪ {as}, so Hec = (Hc \ {as})∪ {hr} (see [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Example of F(He) and γ2: as moves to the bottom row and is removed by (2) in the construction of Γ, while the entry hr is lifted to the second row and is removed by applying δ2p−1 We now study the signs of γ1 and γ2 in the expression of δ2p−1(Γ). Assume first that as < hr (see [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: ), and F = [1, 1 ′ , 2 ′ , 3 ′ ]. Hence δ3(F) = [1′ , 2 ′ , 3 ′ ] − [1, 2 ′ , 3 ′ ] + [1, 1 ′ , 3 ′ ] − [1, 1 ′ , 2 ′ ] [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: ) • 4 ′ is adjacent to 1, but not to 1′ , 2 ′ , 3 ′ ; • 2 is adjacent to 1′ , 2 ′ , 3 ′ , but not to 1; • the facets [1, 1 ′ , 2 ′ , 4 ′ ], [1, 1 ′ , 3 ′ , 4 ′ ], [1, 2 ′ , 3 ′ , 4 ′ ], [2, 1 ′ , 2 ′ , 3 ′ ] belong to supp(Γ). Moreover, {2, 4 ′} ∈/ E(∆), otherwise the induced subgraph on W′ = {1, 2} ∪ {1 ′ , 2 ′ , 3 ′ , 4 ′} would be isomorphic to K2,4 [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Running example - Step 3 Step 4. By duality, for every 1 ≤ h ≤ j, the facet [1, . . . , bh, . . . , j, j + 1, 1 ′ , . . . ,(2p−j) ′ ] belongs to supp(Γ). Hence [1, . . . , bh, . . . , j, j + 1, 1 ′ , . . . , kb′ , . . . ,(2p − j) ′ ] appears in δ2p−1(Γ) for all 1 ≤ k ≤ 2p − j. Since δ2p−1(Γ) = 0 and {j + 1,(2p−j + 1)′} ∈/ E(∆), these terms must cancel with terms coming from new facets in supp(Γ). Therefor… view at source ↗
Figure 11
Figure 11. Figure 11: Running example - Step 5 (Final) Step 5. Observe that [1, . . . , bh, . . . , j, j + 1, 1 ′ , . . . , kb′ , . . . ,(2p − j) ′ ,(2p − j + 1 + h) ′ ] ∈ supp(Γ). Moreover, from the construction developed so far, neither {j + 1+h,(2p−j + 1+k) ′} nor {h,(2p−j+1+k) ′} is an edge of ∆. Therefore, representing [1, . . . , bh, . . . , j, j+1, 1 ′ , . . . , kb′ , . . . ,(2p− j) ′ ,(2p−j + 1 +h) ′ ] a facet in F(∆, … view at source ↗
Figure 12
Figure 12. Figure 12: Running example - Step 7 assume that {5, 5 ′} is not an edge of ∆ (while {3, 5 ′} and {4, 5 ′} are edges, although this is not essential). Then [2, 1 ′ , 2 ′ , 5 ′ ] belongs to the support of Γ, and in this case we may take F1 = {2, 1 ′ , 2 ′ , 5 ′}. The process must then continue by introducing new vertices. For instance, repeating Step 2 for [1′ , 2 ′ , 5 ′ ] yields a new vertex 6 (playing the role of j… view at source ↗
Figure 13
Figure 13. Figure 13: A 1-dimensional flag simplicial complex [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Betti table of I [p] ∆ in the presence of K2+i,2p−i . where β1,2p+2(I [p] ∆ ) = X 0≤i≤p i even [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Betti table of I [p] ∆ in the presence of Cr(2p + 1). where β2p+1, 4p+2(I [p] ∆ ) = p [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Some graphs. It is worth noting that in Example 3.2, the complement of G1 consists of an isolated vertex and the leftmost graph in [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
read the original abstract

Let $G$ be a graph and $I(G)$ its edge ideal. The $p$-th squarefree power $I(G)^{[p]}$ is the monomial ideal generated by squarefree monomials corresponding to the matchings of size $p$ of $G$. In this paper, we provide a combinatorial characterization of when $I(G)^{[p]}$ is linearly related, i.e., when its first syzygy module is generated by linear forms. Moreover, for a $1$-dimensional flag simplicial complex $\Delta$ and its Stanley-Reisner ideal $I_{\Delta}$, which arises as the edge ideal of the complement graph of $\Delta$, we describe the shape of the Betti table of $I_{\Delta}^{[p]}$ and we give a combinatorial characterization of when $I_{\Delta}^{[p]}$ has a linear resolution.

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

0 major / 2 minor

Summary. The paper claims to provide a combinatorial characterization of when the p-th squarefree power I(G)^{[p]} of the edge ideal of a graph G is linearly related (i.e., its first syzygy module is generated by linear forms). For a 1-dimensional flag simplicial complex Δ, it additionally describes the shape of the Betti table of the squarefree power I_Δ^{[p]} of the associated Stanley-Reisner ideal and gives a combinatorial characterization of when this ideal has a linear resolution.

Significance. If the claimed equivalences hold, the results supply explicit graph-theoretic criteria (in terms of matchings of size p and properties of flag complexes) that determine homological properties of monomial ideals. This strengthens the dictionary between combinatorial structures and syzygies, potentially enabling direct verification of linear relatedness or linear resolutions without computing minimal free resolutions algebraically.

minor comments (2)
  1. The abstract states that I_Δ arises as the edge ideal of the complement graph of Δ; a brief sentence in the introduction recalling this standard correspondence would improve accessibility for readers less familiar with Stanley-Reisner theory.
  2. Notation for squarefree powers is consistent, but the manuscript would benefit from an explicit reminder (perhaps in §2) that the generators of I(G)^{[p]} are precisely the squarefree monomials corresponding to matchings of size p.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, the assessment of its significance, and the recommendation to accept. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes combinatorial characterizations (in terms of matchings of size p and properties of 1-dimensional flag complexes) that are proven equivalent to the algebraic conditions of linear relatedness and linear resolution for squarefree powers of edge ideals. These equivalences are the content of the theorems themselves rather than inputs; no step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation. The derivation chain is self-contained via direct combinatorial-algebraic arguments.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or invented entities; all content is stated at the level of standard definitions in commutative algebra.

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