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Trees whose path ideals have linear quotients

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Pith's one-line read Tree n-path ideals are linear exactly when a finite obstruction list is absent

desk verdict A clean and likely-true classification of trees whose n-path ideals have linear quotients, but the proof currently rests on a gap in the trimming lemma that looks repairable. read the letter →

arxiv 2506.06209 v1 pith:GLQCRRI2 submitted 2025-06-06 math.AC math.CO

Trung Chau , Kanoy Kumar Das , Animikha Dutta Dhar , Pranath S Karanth , Aniruda Suswaram This is my paper
classification math.ACmath.CO MSC 13F5505E4005C05
keywords pathidealslinearquotientsresolutiontreescaterpillartrimmingoperationforbiddeninducedsubgraphsmonomial
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

The paper classifies, for every n≥4, which trees have n-path ideals with linear quotients. Its main theorem states that for a tree G, the ideal J_n(G) has linear quotients if and only if it has linear resolution, and both are equivalent to G containing none of a short list of forbidden induced subgraphs: P_n+P_n, plus the graphs L_{n,k} for k in [3,(n+1)/2], with the extra L_{5,3} when n=4. The result matters because linear quotients is a concrete, order-dependent property that is generally harder to certify than linear resolution; the equivalence turns both into a finite graph-checking problem. The proof's engine is a 'trimming' operation that replaces G by a caterpillar tree carrying the same path ideal whenever the obstructions are absent, after which linear quotients are exhibited through an explicit lexicographic order on generators.

What carries the argument

The load-bearing mechanism is the trimming operation. For a tree satisfying condition (F_n)—diameter between n−1 and 2n−1 and none of the forbidden induced subgraphs—trim(G) is the induced subgraph on the closed neighbourhoods of the vertices of a longest path. The key reduction, Lemma 4.5, proves J_n(G)=J_n(trim(G)); trim(G) is a caterpillar tree, so the whole problem reduces to path ideals of caterpillar trees, where the minimal generators have an explicit form. The proof of linear quotients then proceeds by ordering those generators with a carefully chosen lexicographic order and checking that every colon ideal is generated by variables.

What would settle it

Search for a tree G satisfying condition (F_n) together with an n-vertex path having at least one vertex outside the closed neighbourhoods of a chosen diameter path; such a tree would disprove Lemma 4.5, and a computer search over small trees could look for it. A direct computation of reg(J_4(L_{5,3})) or reg(J_n(L_{n,k})) would instead test the regularity estimates in Lemmas 3.3 and 3.4.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 5.1: for a tree G and n≥4, the conditions 'J_n(G) has linear quotients', 'J_n(G) has linear resolution', and 'G avoids the forbidden induced subgraphs' coincide. The forbidden list is P_n+P_n for all n≥4, L_{n,k} for k∈[3,(n+1)/2] when n≥5, and additionally L_{5,3} when n=4. The equivalence is established by showing that in the absence of these subgraphs, G satisfies condition (F_n), which forces J_n(G)=J_n(trim(G)) with trim(G) a caterpillar tree; the trimmed ideal is then shown to have linear quotients by a lexicographic ordering of its minimal generators.

Load-bearing premise

The proof depends on Lemma 4.5, which asserts that under condition (F_n) every n-vertex path of G lies inside trim(G); if that statement fails for some tree, the reduction to caterpillar trees—and with it the implication from the forbidden-subgraph condition to linear quotients—does not go through.

Editorial extensions

If this is right

  • For any fixed n≥4, deciding whether a tree's n-path ideal has linear quotients or linear resolution becomes a finite check for induced P_n+P_n and L_{n,k}.
  • The trimming lemma gives an explicit caterpillar tree that computes the path ideal, so generators of J_n(G) can be listed directly for such trees.
  • Since linear quotients implies linear resolution and the theorem gives the reverse implication for trees, the two properties never diverge for n≥4 tree path ideals.
  • The classification extends the known n=2 and n=3 results (edge ideals and connected ideals) to every n, with the single exceptional obstruction L_{5,3} appearing only at n=4.

Reading between the lines

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

  • One could test whether the same obstruction list characterizes linear quotients for graphs beyond trees, such as chordal graphs or graphs with a single cycle, using the trimming construction as a starting point.
  • The trimming idea may transfer to path ideals of directed graphs or to connected ideals, since the phenomenon that adding edges does not always add generators appears there too.
  • The special role of L_{5,3} for n=4 suggests that exceptional small obstructions may arise in other families, and the L_{n,k} family could have further members relevant to other values of n.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper studies n-path ideals J_n(G) of trees. For n≥4 it claims a complete classification: J_n(G) has linear quotients if and only if it has linear resolution if and only if G avoids certain induced subgraphs, namely P_n+P_n and L_{n,k} for 3≤k≤(n+1)/2, with the additional exception L_{5,3} for n=4. The proof introduces a trimming operation that, under the avoidance condition, replaces any tree by a caterpillar tree with the same path ideal, and then establishes linear quotients via an explicit lexicographic order on the minimal generators. The necessity direction is shown by regularity computations using Eliahou-Kervaire splittings.

Significance. If the proof is completed, the result would give a clean combinatorial characterization linking linear quotients and linear resolution for path ideals of trees, extending the known cases n=2,3. The trimming reduction is an interesting idea that may be useful beyond this paper. The paper is largely self-contained and the computations with Eliahou-Kervaire splittings are explicit and checkable. The main caveat is that the trimming lemma, which is the key reduction, currently has a gap in one subcase of its proof.

major comments (1)
  1. [Lemma 4.5, Case 2 (Section 4)] The proof of Lemma 4.5, Case 2, asserts that the induced subgraph on the vertex set {z_1,...,z_n} ∪ {w_{s-u+1},...,w_{s-1}} is isomorphic to L_{n,u}. This is not established. Claim 4.6 only proves that the path z_1,...,z_n intersects the diameter path, and after setting w_s=z_u the proof derives u≤s≤(d+2)/2. These inequalities do not determine the direction in which the z-path continues after z_u. If z_{u+1}=w_{s-1}, then the w-vertices in the displayed set already belong to the z-path, and the induced subgraph is a path (or a path with a pendant), not L_{n,u}. For example, with n=6, d=6, s=4, u=4, and z=(z_1,z_2,z_3,w_4,w_3,w_2), all conditions before Case 2 are satisfied and the displayed set induces a 7-vertex path. The proof gives no argument excluding this subcase; it may be that this configuration is forbidden by a different induced L_{n,k}, but the proof does not exhibit one. Since Lemma 4.5 is the only step that reduces an arbitrary tree satisfying (F_n) to a caterpillar tree in the proof of Theorem 5.1, this gap is load-bearing. The subcase appears to be repairable, for instance by using the opposite tail {w_{s+1},...,w_{s+u-1}} when the z-path runs backward along the diameter path, but that argument is not in the manuscript.
minor comments (4)
  1. [Section 3, Lemma 3.2] The displayed vertex set "y_{2k−n}, y_{n−k+1}, . . . , y_{k−1}" does not define the intended segment; the correct set for the claimed isomorphism is {y_{2k−n},...,y_{k−1}} (followed by reversing the x-path). As printed, the argument is hard to follow.
  2. [Section 5, Propositions 5.4 and 5.5] The subscript "i−n+3" in "x_{i−n+3,k}" is a typo for "i+n−3"; the same typo appears in subcase (c) of Proposition 5.5. This makes the containment (m':m) ⊆ (...) appear to refer to nonexistent variables.
  3. [Section 5, proof of Theorem 5.1] The notation "LN_G(x_{n−2}) = LN_G(x_{n+1}) = 0" should read "= ∅" (or "is empty"), since the left-hand side is a set.
  4. [Introduction, paragraph on path ideals] The statement "J_4(G) = (0) for any star graph G" should specify that the star must have at least n vertices; otherwise, when the vertex count is smaller than n, the statement is vacuously true but the wording may be confusing.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained, using external standard results and explicit combinatorial arguments; the trimming step is graph-theoretic, not defined in terms of J_n(G).

full rationale

The paper's central claims do not reduce to their inputs. The implication (1)⇒(2) uses the external Herzog–Hibi theorem (Lemma 2.1), and (2)⇒(3) uses the external Restriction Lemma (Lemmas 2.2, 2.3) together with explicit regularity computations for P_n+P_n and the newly introduced obstructions L_{n,k} (Lemmas 3.1, 3.3, 3.4). The only potentially load-bearing internal step, Lemma 4.5, is not circular: trim(G) is defined purely graph-theoretically as the induced subgraph on the closed neighborhoods of a diameter path, independently of J_n(G), and the equality J_n(G)=J_n(trim(G)) is then argued combinatorially via branching considerations and forbidden induced subgraphs. The subsequent reduction to caterpillar trees is then verified by explicit lex-order colon-ideal checks in Propositions 5.4–5.6. There are no fitted parameters, no predictions statistically forced by a fit, and no load-bearing self-citations. The only self-citation appears in the introduction, where reference [8] is mentioned as related work on regularity of path ideals of caterpillar graphs; it is not used in the proof of the main theorem. The skeptical concern about a possible gap in Case 2 of Lemma 4.5 is a correctness issue, not a circularity issue: even if that argument were incomplete, the claim would not be assumed or defined into existence. This paper is therefore self-contained against external benchmarks and should receive a circularity score of 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper relies only on standard commutative algebra background and the external caterpillar characterization. No free parameters are fitted, and no new postulated entities are introduced.

assumptions (4)
  • standard math Equigenerated monomial ideals with linear quotients have linear resolution (Lemma 2.1, from [15]).
    Used to prove that (1) implies (2) in Theorem 5.1.
  • standard math Restriction lemmas for linear resolution and linear quotients (Lemmas 2.2 and 2.3).
    Used to show forbidden induced subgraphs obstruct the desired algebraic properties.
  • standard math Eliahou-Kervaire splitting regularity formulas (Lemma 2.5 and [11, Corollary 2.7]).
    Used to compute regularity of J_n on small graphs such as P_n+P_n and L_{n,k}.
  • domain assumption Harary-Schwenk theorem: a tree without the subdivided claw L_{5,3} as an induced subgraph is a caterpillar.
    Used in Lemma 4.2 and Lemma 4.5 for n=4,5 to reduce to caterpillar trees.

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Pith. "Pith review of Trees whose path ideals have linear quotients." pith.science (2026). https://pith.science/paper/GLQCRRI2

@misc{pith2026250606209,
  author       = {Pith},
  title        = {Pith review of: Trees whose path ideals have linear quotients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GLQCRRI2}},
  note         = {Machine review of arXiv:2506.06209}
}
abstract

For any integer $n$, we classify all trees whose $n$-path ideals have linear quotients.

Figures

Figures reproduced from arXiv: 2506.06209 by the authors.

Figure 1
Figure 1. The graph G forbidden structures in a tree. Section 4 discusses the trimming operation, and shows that in the absence of the forbidden structures, the trimming operation depends only on the original graph. In Section 5, we prove our main result. Acknowledgements Chau and Das are partially supported by a grant from the Infosys Foundation. 2. Preliminaries In this section, we recall several notions from graph theory a… view at source ↗
Figure 2
Figure 2. The graph Ln,k The graph Ln,k is a tree. If k ∈ [(n + 1)/2, n − 2], it is straightforward that the induced subgraph of Ln,k on the vertex set x1, . . . , xn, y2k−n, yn−k+1, . . . , yk−1 is isomorphic to Ln,n−k+1 where n − k + 1 ∈ [3,(n + 1)/2]. This implies the following result. Lemma 3.2. A graph G does not contain Ln,k, where k ∈ [3,(n+ 1)/2], as an induced subgraph if and only if it does not contain Ln,k, where k… view at source ↗
Figure 3
Figure 3. A caterpillar graph G [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Ld+1,a is induced by {zi , wi : i ∈ [d + 1]} It is clear that this graph is isomorphic to Ld+1,a, where a ∈ [3, d − 1]. Since d + 1 ≥ n (due to the (Fn) condition), it contains Ln,a as an induced subgraph. This contradicts the fact that G satisfies the (Fn) condition, …
Figure 5
Figure 5. Figure 5: The subgraph induced by the vertex set {zi , wj : i ∈ [n], j ∈ [d+ 1]} It is clear that this graph is isomorphic to Ln,r where we already know that r ∈ [3,(n+ 1)/2]. This contradicts the assumption that G satisfies the (Fn) condition. Next, we assume that r ≤ 2. Then, …
Figure 6
Figure 6. Figure 6: The subgraph induced by the vertex set {zi , wj : i ∈ [n], j ∈ [d+ 1]} It is clear that this graph contains a induced subgraph isomorphic to Ln,t for some t ∈ [3, n − 2]. This contradicts the assumption that G satisfies the (Fn) condition. □ By the above claim, the ind…
Figure 7
Figure 7. Figure 7: The subgraph induced by the vertex set {zi , wj : i ∈ [n], j ∈ [d+ 1]} The indices make sense, since s + 3 ≤ n + 3 = n + 6 − 3 ≤ n + (n) − 3 = 2n − 3 ≤ d + 1, s − n + 4 = (s − n + 1) + 3 ≥ 0 + 3 ≥ 1, u − n + 4 = (u − n + 1) + 3 ≥ 0 + 3 ≥ 1. This graph is isomorphic to …
Figure 8
Figure 8. Figure 8: The subgraph induced by the vertex set {zi , wj : i ∈ [n], j ∈ [d+ 1]} The indices are clearly well-defined. This graph is isomorphic to Ln,u where u ∈ [3, n − 2]. This contradicts the assumption that G satisfies the (Fn) condition, as desired. This concludes the proof…

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