On induced cycles of Levi graphs associated to line arrangements

Rajib Sarkar; Rupam Karmakar

REVIEW 5 major objections 4 minor 11 references

On induced cycles of Levi graphs associated to line arrangements

T0 review · 5 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that Levi graphs of line arrangements contain induced cycles of explicitly bounded even lengths growing linearly with the number of lines, and it computes the longest induced cycle exactly for several special arrangements.

desk verdict Real content in the specific computations, but the paper's main general theorem is not proved as stated; worth a revise-and-resubmit, not a desk reject. read the letter →

arxiv 2411.18488 v1 pith:ZL3RZIUV submitted 2024-11-27 math.CO math.AG

Rupam Karmakar , Rajib Sarkar This is my paper · ORCID

classification math.COmath.AG MSC 14N1014N2005C3805C1005E14

keywords linearrangementsLevigraphsinducedcyclesbipartitesupersolvableHessearrangementCevalongestcycle

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 studies induced cycles in Levi graphs, the bipartite graphs recording incidences between lines of a line arrangement and their intersection points. Its main results show that, under mild restrictions on intersection multiplicities, these graphs must contain induced cycles whose length grows linearly with the number of lines. For arrangements whose points are only double or triple points, an induced cycle of every even length $2i$ exists up to about $k/2$ lines when $k$ is odd, with a general bound extending to maximum multiplicity $q$. The paper also determines exact longest induced cycles for the Hesse arrangement (length 12), a $(9_3)$ arrangement (length 14), and certain supersolvable arrangements (length $4m$).

What carries the argument

The carrying mechanism is a greedy line-by-line construction of an induced cycle. The proof selects a sequence of lines $\ell_{j_1},\dots,\ell_{j_i}$ so that each new line meets its predecessor at a point lying on no earlier chosen line, and then counts the available choices at step $j$ as $k$ minus the number of lines forbidden by earlier intersections; the multiplicity restrictions keep this count positive up to the claimed length. The central object is the Levi graph, whose vertices are lines and intersection points, with an induced cycle corresponding to selected intersection points that lie on no line other than their two neighbouring lines in the cycle.

What would settle it

Take the Hesse arrangement's Levi graph (12 line vertices and 21 point vertices) and run an exhaustive search for an induced cycle on 14 vertices; the paper claims no such cycle exists. Alternatively, for an arrangement satisfying the hypotheses of Theorem 4.6(iii), explicitly list the forbidden lines at the final step of the construction and check that the count stays at least 1; a violation would show the proof does not establish the stated range.

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Extended reading notes

Core claim

The central claim is that the Levi graph of a line arrangement contains induced cycles of explicitly bounded even length whenever the arrangement's intersection points have controlled multiplicity. Theorem 4.9 states that if no point has multiplicity exceeding $q$, then induced cycles of length $2i$ exist for every $i \le \lfloor (k+9q-18)/(3q-5)\rfloor$; for the double-and-triple-point case $q=3$, this gives cycles of all even lengths up to about half the number of lines. Beyond this existence result, the paper computes the length of the longest induced cycle exactly for the Hesse arrangement (12), a $(9_3)$ arrangement (14), and supersolvable arrangements of type $A(w,k)$ with $k=2,3,4$ ($4m$).

Load-bearing premise

The counting argument assumes that at every step the sets of lines forbidden by previously chosen intersection points overlap as little as possible, so that the total number of forbidden lines never exceeds the stated linear bound.

Editorial extensions

If this is right

For line arrangements with only double and triple points, the Levi graph is guaranteed to contain induced cycles whose length is at least about $k/2$ when $k$ is odd.
For arrangements with maximum intersection multiplicity $q$, induced cycles of length about $2k/(3q)$ are guaranteed to exist.
The exact longest-cycle values for the Hesse, $(9_3)$, and some supersolvable arrangements show that the true longest induced cycle can be significantly shorter than the general lower bound suggests.
Since the paper's motivation is algebraic, these induced-cycle results give lower bounds for the Castelnuovo–Mumford regularity of powers of binomial edge ideals of the corresponding Levi graphs.
For supersolvable arrangements, no induced cycle can use all $2k$ vertices, so the Levi graph is never Hamiltonian in the induced sense.

Reading between the lines

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

The greedy counting method may be improvable: replacing worst-case overlap assumptions with a double-counting of forbidden lines could tighten the constants in Theorems 4.6 and 4.9 without changing the overall linear growth.
The exact longest-cycle computations for the Hesse, $(9_3)$, and supersolvable families suggest that a general upper bound for induced cycles might be governed by the arrangement's modular points or by its largest multiplicity, a connection the paper does not fully explore.
The algebraic translation to binomial edge ideal regularity could be made quantitative: if the induced-cycle bounds are optimal, they would pin down the regularity growth for these ideals, but the paper does not compute the upper bounds needed for that conclusion.
A computational survey of small line arrangements could test whether the theorem's ranges are tight, for instance whether any arrangement with only double and triple points achieves exactly the bound $\lfloor (k+9)/4\rfloor$.

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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

Here's the short version: this paper has some real content, but its main general theorem (4.9) is not proved as stated, and there are a couple of algebraic typos that need fixing. The specific computations for Hesse and supersolvable arrangements are the strongest part.

What's actually new: the C8 existence result under tk=tk-1=0 (Theorem 4.2) is neat, with a sharp near-pencil example. The explicit longest-cycle computations for the (9_3), the 10-line extension, Hesse (12), and supersolvable families (8 for μ=4, 2(2m−2) for μ=3, 4m for k=2,3,4 variants) are concrete and checkable. These are useful data points for a subfield that mostly has existence bounds.

Where it gets soft: Theorem 4.9(i) is the main general claim, stated for arrangements with t_q≠0 and t_r=0 for r>q (so lower multiplicities are allowed). The proof starts 'Let us assume that Sing L has only q-fold points' and never returns to the general case. That's a missing reduction. It's plausibly fixable—lower-multiplicity points forbid fewer lines in the greedy count, so the q-fold case may indeed be the worst case—but the paper doesn't argue it. That is a genuine gap in a central result.

Theorem 4.6 also has typos: in part (iii) the final inequality drops the '/2' from (7i−18)/2, and part (i) substitutes ⌊(k+5)/3⌋ where the theorem needs ⌊(k+9)/4⌋. These look like slips rather than fatal errors; the intended inequalities do hold.

The upper-bound case analyses for specific arrangements (e.g., Hesse, Example 4.7) are long and sometimes hand-wavy ('the proofs are similar, details omitted'). They're probably right, but they'd need careful checking.

The citation pattern is fine. The prior C6 theorem is the same group's earlier published work, used as a starting point, legitimately.

Bottom line: this is honest work, but in its current form it's not quite there. It deserves a serious referee, not a desk rejection. A competent referee will catch the Theorem 4.9 gap and the typos, and the authors should be able to fix them. I'd give it a 'revise and resubmit' recommendation, and I'd want to see whether the general theorem can be repaired or needs to be weakened.

Referee Report

5 major / 4 minor

Summary. The paper studies induced even cycles in Levi graphs associated to complex line arrangements. It proves existence of induced C_8 when t_k = t_{k-1} = 0, gives criteria for induced C_10 under large-multiplicity hypotheses, and then states general existence bounds for arrangements with maximal multiplicity q: Theorem 4.6 for q = 3 and Theorem 4.9 for general q. It also computes or bounds longest induced cycles for several explicit arrangements: the (9_3) arrangement (length 14), one 10-line extension (length 18), the Hesse arrangement (length 12), Ceva arrangements, and several supersolvable families, including a claimed maximum length 4m for the family A(w,k) with k = 2,3,4. The proofs are constructive and rely on explicit line selections and case analyses.

Significance. If the existence bounds are correct, they provide the first general lower bounds of this type for induced cycles of length at least 8 in Levi graphs of line arrangements, and the explicit longest-cycle computations for Hesse and supersolvable families are useful data points. The connection to binomial edge ideals mentioned in the introduction gives the results potential algebraic relevance. The paper is clearly written in structure and the explicit cycles in Examples 4.7, 4.8, and Theorem 4.10 are checkable. However, the main general theorem, Theorem 4.9, has a serious proof gap: the proof assumes a stronger hypothesis than the statement, and the stated range is not derived for the cases actually covered. Several further counting arguments contain algebraic slips or informal accounting. The central claims are therefore plausible but are not established as written.

major comments (5)

[Theorem 4.9(i), proof] The theorem is stated for arrangements with t_q ≠ 0 and t_r = 0 for all r > q, which allows lower-multiplicity points. The proof, however, begins with the extra hypothesis 'Sing L has only q-fold points i.e. t_r = 0 for all r < q' and never explains how to reduce the stated case to that homogeneous case. All subsequent counts, such as k − {(i−1) + [(i−2)+(i−3)+(i−4)](q−2)} choices for ℓ_ji, are derived in the all-q-fold setting. In the presence of double or other lower-multiplicity points, the forbidden sets have different sizes and different overlaps, so those counts do not apply. Since Theorem 4.9 is the paper's most general existence result, this is a load-bearing gap; the stated range i ≤ ⌊(k+9q−18)/(3q−5)⌋ is not proved for the arrangements covered by the theorem as stated.
[Theorem 4.9(ii), proof] The proof does not establish the bound stated in the theorem. In Case I, the count k − {(i−1)+2(p−2)+(3i−11)(q−2)} leads to the bound i ≤ ⌊(k−2p+11q−18)/(3q−5)⌋, and in Case II the bound is of the form ⌊2(k+p+8q−18)/(p+5q−10)⌋. Both depend on p, whereas the theorem claims the p-independent bound i ≤ ⌊(k+10q−18)/(3q−5)⌋. No argument shows that the p-dependent bounds imply the stronger p-independent one; indeed for p close to q the stated bound is larger than the Case I bound. The proof must either derive the stated range or the theorem's statement must be adjusted.
[Theorem 4.16(iii), proof, k = 3 and k = 4] The proof for k = 3 concludes 'maximum length of an induced cycle in G3 is ≤ 2m' and then 'maximum length of an induced cycle in G3 is 2 m'; similarly for k = 4 it concludes a maximum of 2m. This contradicts the theorem's assertion that the maximum length is 4m. No argument is given that rules out induced cycles longer than 2m, nor is it explained how 4m could be the maximum if the proof shows a bound of 2m. This inconsistency affects the claimed exact values for the longest induced cycles of A(w,3) and A(w,4) and must be resolved.
[Theorem 4.6(iii), final inequality] The proof derives that there are at least k − (7i−18)/2 choices for ℓ_ji, but the final inequality is written as k − (7⌊(2k+16)/7⌋ − 18) ≥ 1, losing the division by 2. The displayed inequality does not follow from the preceding count. The intended bound can be repaired using the correct expression, but as printed the proof of part (iii) is invalid at the final step.
[Theorems 4.6, 4.9, 4.16, counting arguments] The greedy counting arguments are informal: statements such as 'it might happen', 'we have to further remove i−3 more choices', and 'we remove i−4 choices' are not accompanied by a precise accounting of which lines are forbidden at each step and why the forbidden sets overlap in the claimed way. Since the existence ranges in Theorems 4.6, 4.9, and 4.16 depend on these counts, the argument needs to be formalized, for instance by specifying the exact set of candidate lines after each step and proving an upper bound on the number of forbidden candidates that is monotone in the step index.

minor comments (4)

[Theorem 4.10, proof of Claim III] The sentence 'For other choices of (j1, j2)proo' is truncated, and the next paragraph does not complete the case analysis. The proof should finish this sentence and spell out the remaining choices of (j1,j2), or state explicitly that they are symmetric and indicate why.
[Theorem 4.16(i), proof] The proof contains a likely typo: it says 'we claim that the maximum length of an induced cycle in G0 is 2(m−2)', although the theorem asserts 2(2m−2) and the preceding construction gives a cycle of length 2(2m−2). The argument that follows rules out a cycle of length 2(2m−1), so the intended claim is evidently 2(2m−2). The two occurrences '2(m−2)' and '2(m−1)' should be corrected.
[Theorem 4.10, proof] The proof of Claim III contains a stray period in 'Now, let S has at most two double points. .' and the logical transition from Claims I–III to the final contradiction for i = 7 and i ≥ 8 would be easier to follow if the cases were labeled explicitly.
[Throughout] There are several minor typos and grammatical slips, e.g., 'integar' in Example 4.4, 'doesn’t contain induced cycles' in the introduction, and inconsistent spacing around floor expressions. These do not affect the mathematics, but a careful copyedit is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: results are proved by explicit cycle constructions and counting arguments; the only self-citation is not load-bearing.

full rationale

The paper's main claims are established by direct constructive arguments. Theorem 4.2 explicitly builds an induced C8, Theorem 4.5 constructs induced C10s case by case, and Theorems 4.6, 4.9, 4.11, 4.15, and 4.16 count available choices for the next line of a cycle under construction rather than assuming the desired cycle already exists. The upper-bound proofs for Examples 4.7 and 4.8 and Theorems 4.10, 4.14, and 4.16 use contradiction arguments starting from a hypothetical longer cycle, so they do not reduce to their conclusions. The only citation to the authors' own prior work is [10, Theorem 5.1] for the existence of an induced C6; it is used as motivation and context, and none of the new bounds depend on it. The classifications cited in Theorems 4.13–4.16 are from Hanumanthu–Harbourne and Abe–Dimca, not from the present authors. There are genuine proof gaps: Theorem 4.9(i) begins with the extra hypothesis 'Let us assume that Sing L has only q-fold points,' which is stronger than the stated hypothesis, and Theorem 4.6(iii) contains an algebraic slip in the final inequality. These are rigor concerns, not circularity, because they do not make the claimed conclusions inputs to the derivations. Under the requested standard, no specific circular reduction can be exhibited from the paper's own equations.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The results are pure combinatorial theorems; no free parameters are fitted and no new entities are introduced. The proofs rely on standard combinatorial identities and two external classification theorems.

assumptions (2)

standard math Combinatorial counts for line arrangements: C(k,2) = sum_p C(m_p,2) and k-1 = sum_{p on line} (m_p-1).
Used in Section 4 to relate multiplicities and in parity arguments like 'each line has at least one double point when k is even and only double/triple points exist.' These are standard identities, but the parity consequence is invoked without proof.
domain assumption Classification of supersolvable line arrangements (Theorem 4.13).
Imported from [7] and [1]; it describes all supersolvable line arrangements with given modular-point data. Used to set up Theorems 4.14, 4.15, and 4.16.

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Pith. "Pith review of On induced cycles of Levi graphs associated to line arrangements." pith.science (2026). https://pith.science/paper/ZL3RZIUV

@misc{pith2026241118488,
  author       = {Pith},
  title        = {Pith review of: On induced cycles of Levi graphs associated to line arrangements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZL3RZIUV}},
  note         = {Machine review of arXiv:2411.18488}
}

abstract

In this article, we investigate the existence of induced cycles in Levi graphs associated to line arrangements in $\mathbb{P}_{\mathbb{C}}^2$. We also look at the problem of finding the length of a longest induced cycle in Levi graphs associated to line arrangements.

Figures

Figures reproduced from arXiv: 2411.18488 by the authors.

**Figure 2.** Levi Graph G = C2(2i+4) associated to the supersolvable line arrangement L We show that G contains an induced cycle of length 2(2i+ 3) for every 1 ≤ i ≤ m−3. For i = 1, we consider the lines ℓ1 = L 0 xy, ℓ2 = Lz, ℓ3 = L 1 xy, ℓ4 = L 0 xz, ℓ5 = L 1 yz. In addition, we consider the corresponding intersection points p12 = (1, 1, 0), p23 = (ǫ, 1, 0), p34 = (ǫ, 1, ǫ), p4,5 = (1, ǫ, 1) and p15 = (ǫ, ǫ, 1). Clearly, pii+1… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

On complex supersolvable line arr angements

Takuro Abe and Alexandru Dimca. On complex supersolvable line arr angements. J. Algebra, 552:38–51, 2020

work page 2020

[2] [2]

Ore and Erd˝ os type condition s for long cycles in balanced bipartite graphs

Janusz Adamus and Lech Adamus. Ore and Erd˝ os type condition s for long cycles in balanced bipartite graphs. Discrete Math. Theor. Comput. Sci. , 11(2):57–69, 2009

work page 2009

[3] [3]

Edge condition for long cycles in bipartite graphs

Lech Adamus. Edge condition for long cycles in bipartite graphs. Discrete Math. Theor. Comput. Sci. , 11(2):25–32, 2009

work page 2009

[4] [4]

Cardoso, Marcin Kami´ nski, and Vadim Lozin

Domingos M. Cardoso, Marcin Kami´ nski, and Vadim Lozin. Maximum k-regular induced subgraphs. J. Comb. Optim. , 14(4):455–463, 2007

work page 2007

[5] [5]

H. S. M. Coxeter. Self-dual conﬁgurations and regular graphs . Bull. Amer. Math. Soc. , 56:413–455, 1950

work page 1950

[6] [6]

On some of my favourite problems in various branches of combinatorics

Paul Erd˝ os. On some of my favourite problems in various branches of combinatorics. In Fourth Czechoslo- vakian Symposium on Combinatorics, Graphs and Complexity ( Prachatice, 1990) , volume 51 of Ann. Discrete Math., pages 69–79. North-Holland, Amsterdam, 1992

work page 1990

[7] [7]

Real and complex sup ersolvable line arrangements in the projective plane

Krishna Hanumanthu and Brian Harbourne. Real and complex sup ersolvable line arrangements in the projective plane. J. Algebraic Combin. , 54(3):767–785, 2021

work page 2021

[8] [8]

Cycles in bipartite graphs

Bill Jackson. Cycles in bipartite graphs. J. Combin. Theory Ser. B , 30(3):332–342, 1981

work page 1981