Critical edge sets in vertex-critical graphs

Ema Skottova; Raphael Steiner

REVIEW 3 major objections 3 minor 40 references

Critical edge sets in vertex-critical graphs

T0 review · 3 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For k ≥ 5, vertex-critical graphs can hide all small critical edge sets, the paper proves lower and upper bounds

desk verdict Can't judge the math: the record has only the abstract of the graph theory paper, and the attached full text is an unrelated medical-imaging paper. read the letter →

arxiv 2508.08703 v1 pith:XULY7ZTP submitted 2025-08-12 math.CO

Ema Skottova , Raphael Steiner This is my paper · ORCID

classification math.CO MSC 05C1505C35

keywords vertex-criticalgraphscriticaledgesetschromaticnumberregularitylemmagluingconstructionasymptoticboundsgraphcoloring

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

Criticality distinguishes graphs whose chromatic number drops when a single vertex is removed from those that drop only when an edge is removed. A problem posed in 1985 asked whether, for fixed $k\ge 4$, there are $k$-vertex-critical graphs of order $n$ in which no set of at most $f_k(n)$ edges is critical, with $f_k(n)\to\infty$. This paper answers yes for every $k>4$: it proves $f_k(n)=\Omega(n^{1/3})$, a stronger lower bound of order $\sqrt{n}$ along infinitely many $n$, and, for every $k\ge 4$, the first non-trivial upper bound $f_k(n)=O(n/(\log n)^{\Omega(1)})$. Only the case $k=4$ is left open. The lower-bound proof combines a gluing operation with an exhaustive analysis of proper colorings of a modified known construction; the upper bound follows from a variant of the regularity lemma.

What carries the argument

The gluing operation is the load-bearing device for the lower bound: it combines two $k$-vertex-critical graphs without small critical edge sets into a larger one, provided every proper colouring of the modified construction is compatible with the gluing. An exhaustive enumeration of the proper colourings of that modified example rules out colouring behaviours that would reintroduce a small critical edge set. For the upper bound, the regularity lemma variant decomposes an arbitrary $k$-vertex-critical graph into a bounded-complexity core plus a quasirandom remainder, which locates a critical edge set of size $O(n/(\log n)^{\Omega(1)})$.

What would settle it

For a fixed $k\ge 5$, exhibit arbitrarily large $k$-vertex-critical graphs of order $n$ in which every set of up to $n^{1/3-\varepsilon}$ edges is critical; this would refute $f_k(n)=\Omega(n^{1/3})$. Alternatively, for some $k\ge 4$ find a $k$-vertex-critical graph of order $n$ whose smallest critical edge set has size $\gg n/(\log n)^C$ for every fixed $C$, contradicting the upper bound.

Watch

Extended reading notes

Core claim

The paper's central claim is that the functions $f_k(n)$ grow like a power of $n$ for all $k\ge 5$: every sufficiently large $k$-vertex-critical graph can be chosen so that no set of at most $cn^{1/3}$ edges is critical, for some absolute $c>0$. It further shows $f_k(n)=\Omega(\sqrt{n})$ along an infinite sequence of orders, and proves the complementary bound $f_k(n)=O(n/(\log n)^{\Omega(1)})$ for every $k\ge 4$. The proof of the lower bound is constructive: a modification of an earlier example, analysed colouring-by-colouring, is combined with a gluing operation that preserves vertex-criticality, chromatic number $k$, and the absence of small critical edge sets. The upper bound uses a varia

Load-bearing premise

The lower bound holds only if the gluing operation and the complete case analysis of the modified construction's proper colourings cover every colouring that can arise; if some colouring behaviour escapes the analysis, a small critical edge set could enter and the $\Omega(n^{1/3})$ bound would fail.

Editorial extensions

If this is right

For every $k\ge 5$, there are arbitrarily large $k$-vertex-critical graphs whose smallest critical edge set has size at least $c\,n^{1/3}$.
Vertex-criticality and edge-criticality can diverge arbitrarily in the sublinear range: a graph may be critical under vertex deletion while still having no small critical edge set.
The first non-trivial upper bound caps the possible resilience at $O(n/(\log n)^{\Omega(1)})$ for all $k\ge 4$.
An infinite sequence of orders achieves the stronger $\Omega(\sqrt{n})$ lower bound.
The case $k=4$ remains open; all lower-bound results require $k>4$.

Reading between the lines

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

If the gluing operation can be iterated without degrading the order of the graphs, the lower bound might be pushed to $\Theta(n^{1/2})$ for all sufficiently large $n$, not just along a sparse sequence.
The regularity-lemma upper bound suggests that the true maximum of $f_k$ may be determined by quasirandom behaviour; random-like critical graphs are a natural testbed for whether the $n/(\log n)^c$ bound is tight.
The $k=4$ case may be reparable by adding a small gadget to the gluing that forces all colourings into the analysed cases, since the obstruction is the colouring analysis rather than the gluing concept.

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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

The first thing you should know: we do not have the manuscript for arXiv:2508.08703. The full text attached to this record is a completely different paper, an AS-OCT segmentation paper with a different arXiv ID. So any review of the actual proof would be based on the abstract alone.

That qualification matters because what the abstract claims is genuinely significant. The paper purports to solve Erdős's 1985 problem on critical edge sets in vertex-critical graphs for every k > 4, giving f_k(n) = Ω(n^{1/3}), plus a stronger Ω(√n) bound along an infinite sequence of orders, and the first nontrivial upper bound f_k(n) = O(n/(log n)^{Ω(1)}) for all k ≥ 4. If true, that is a major result in chromatic graph theory — a 40-year-old benchmark settled with polynomial lower bounds. The approach described is plausible and consistent with the literature: modifying Jensen's 2002 construction, using a gluing operation to preserve vertex-criticality while avoiding small critical edge sets, and applying the Conlon–Fox regularity lemma for the upper bound. The abstract also honestly notes that k = 4 remains open.

The soft spot is not a mathematical one I can point to — it is that no proof is inspectable. There are no lemmas, no details of the gluing, no case analysis of proper colorings, no verification that the Conlon–Fox lemma applies. The abstract could be entirely correct, or it could hide a subtle gap; I have no way to tell. That is an evidence gap, not a detected error, but it is decisive for any reviewer. I also note the reader's circularity score of 1.0: nothing in the abstract suggests self-referential reasoning, so that concern does not land.

Who is this for? If the result stands, it is for every researcher in extremal and chromatic graph theory. But this particular record is not yet reviewable. My recommendation: get the real manuscript from arXiv, read the relevant sections, and then send it to a serious referee. The claim is important enough that it deserves careful scrutiny — do not desk-reject based on the abstract alone, but do not accept anything without seeing the proof.

Referee Report

3 major / 3 minor

Summary. The submission is identified as arXiv:2508.08703 (math.CO), and its abstract announces results on critical edge sets in k-vertex-critical graphs: for every k > 4, f_k(n) = Ω(n^{1/3}), with a stronger Ω(√n) lower bound along an infinite sequence of n, and a first nontrivial upper bound f_k(n) = O(n/(log n)^{Ω(1)}) for every k ≥ 4. The proof is said to use a modification of Jensen's 2002 construction, a gluing operation, and a variant of Szemerédi's regularity lemma due to Conlon and Fox, with the k = 4 lower bound left open. However, the only full text supplied in the submission record is arXiv:2508.08705v1, an unrelated medical-image segmentation paper on adaptive confidence-wise loss for AS-OCT lens structure segmentation. Consequently, none of the graph-theoretic arguments, lemma statements, or proof details are present in the record, and the central claims cannot be checked.

Significance. If the announced results are correct, they would resolve Erdős's question for every k > 4 and provide the first nontrivial upper bound for the functions f_k, a substantial advance on a long-standing problem. The lower bound is especially significant because it strengthens previous partial results and leaves only k = 4 open. That said, this assessment is conditional: no proof is available to verify. The submission includes no machine-checked proofs, reproducible code, or parameter-free derivations; the only documented content is the abstract. The significance of the mathematical contribution therefore cannot be confirmed from the submitted record.

major comments (3)

[Abstract / Submitted full text] The central claim f_k(n) = Ω(n^{1/3}) for all k > 4 is asserted in the abstract, but the only full text attached to the submission is an unrelated AS-OCT segmentation paper (arXiv:2508.08705v1). There are no theorem statements, lemmas, proofs, or definitions in the record that support the claimed lower bound. This is a load-bearing gap: the result cannot be verified or reproduced.
[Abstract (gluing operation)] The proof of the lower bound is said to combine a modification of Jensen's construction with a gluing operation that preserves k-vertex-criticality and the absence of small critical edge sets. The abstract describes this operation only qualitatively. No argument is provided that the gluing preserves the three required properties simultaneously, nor is there an analysis of all proper colorings of the glued graph. Without these steps, the Ω(n^{1/3}) claim is unsupported.
[Abstract (upper bound)] The upper bound f_k(n) = O(n/(log n)^{Ω(1)}) is stated to follow from a variant of Szemerédi's regularity lemma due to Conlon and Fox. The record contains no statement of this variant, no verification that its hypotheses are satisfied for k-vertex-critical graphs, and no derivation of the bound. As a result, the upper-bound claim is also unverifiable.

minor comments (3)

[Abstract] The abstract says 'for every k ≥ 4' for the upper bound and 'for all k > 4' for the lower bound; this is clear enough, but a sentence explicitly separating the k = 4 status would help readers.
[Abstract] The stronger lower bound of order √n along an infinite sequence of n is announced without indicating how the infinite sequence is generated or how it relates to the n^{1/3} bound. A remark on the underlying construction would be useful.
[References] The abstract cites Jensen (2002) and Conlon and Fox, but the submitted record contains no bibliography. Full references should be supplied with the correct manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity visible in the available abstract; the attached full text is an unrelated AS-OCT paper, so the proof chain cannot be audited but nothing in the record reduces f_k to itself by construction.

full rationale

The only available text from the claimed graph paper is the abstract. The lower bound f_k(n)=Omega(n^{1/3}) is described as following from an intricate analysis of proper colorings of a modification of Jensen's construction plus a gluing operation, and the upper bound is said to use a Conlon-Fox variant of Szemeredi's regularity lemma. These are external ingredients; they are not defined in terms of f_k, and no equation in the abstract fits a parameter and then renames it as a prediction. The submitted full text is arXiv:2508.08705v1, a medical image segmentation paper about Adaptive Confidence-Wise loss, which is unrelated to the claimed combinatorics paper. This means the derivation's key steps are not inspectable in this record, which is a correctness/reproducibility concern, not a circularity finding. Per the instructions, circularity must be exhibited by quoting a specific reduction; none can be exhibited here. Therefore the appropriate finding is no significant circularity.

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

No free parameters are visible from the abstract: f_k(n) is a defined extremal function, and the constants hidden in Ω and O are existential proof constants rather than numbers fitted to data. The main unverified imports are Jensen's construction and the Conlon-Fox regularity lemma variant. The gluing operation and the coloring analysis, which carry the lower-bound proof, cannot be audited without the manuscript body, which the supplied record does not contain.

assumptions (4)

domain assumption Jensen's 2002 result: for every k ≥ 5, there exists a k-vertex-critical graph with no critical edges (Dirac's conjecture for k ≥ 5).
Invoked in the abstract as the basis for the new construction ('a modification of an earlier construction due to Jensen'); its correctness is imported, not re-proved.
domain assumption The Conlon-Fox variant of Szemerédi's regularity lemma is correct and applicable to the graphs used in the upper-bound proof.
The abstract states the upper bound f_k(n) = O(n/(log n)^{Ω(1)}) 'is obtained using a variant of Szemerédi's regularity lemma due to Conlon and Fox'; the lemma and its applicability are assumed.
standard math Standard definitions and elementary facts about k-chromatic, vertex-critical, and edge-critical graphs (Dirac, 1950s).
Background for the definitions of f_k(n) and for the claim that removing a set of edges lowers the chromatic number; standard textbook material.
standard math The extremal function f_k(n) is well-defined, i.e., for each admissible n there exists at least one k-vertex-critical graph of order n whose removal resilience attains the stated extremes.
Needed for the Ω and O bounds to have meaning; standard existence reasoning in chromatic graph theory.

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Cite this review

Pith. "Pith review of Critical edge sets in vertex-critical graphs." pith.science (2026). https://pith.science/paper/XULY7ZTP

@misc{pith2026250808703,
  author       = {Pith},
  title        = {Pith review of: Critical edge sets in vertex-critical graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XULY7ZTP}},
  note         = {Machine review of arXiv:2508.08703}
}

abstract

Criticality is a fundamental notion in graph theory that has been studied continually since its introduction in the early 50s by Dirac. A graph is called $k$-vertex-critical ($k$-edge-critical) if it is $k$-chromatic but removing any vertex (edge) lowers the chromatic number to $k-1$. A set of edges in a graph is called critical if its removal reduces the chromatic number of the graph. In 1970, Dirac conjectured a rather strong distinction between the notions of vertex- and edge-criticality, namely that for every $k\ge 4$ there exists a $k$-vertex-critical graph that does not have any critical edges. This conjecture was proved for $k\ge 5$ by Jensen in 2002 and remains open only for $k=4$. A much stronger version of Dirac's conjecture was proposed by Erd\H{o}s in 1985: Let $k\ge 4$ be fixed, and let $f_k(n)$ denote the largest integer such that there exists a $k$-vertex-critical graph of order $n$ in which no set of at most $f_k(n)$ edges is critical. Is it true that $f_k(n)\rightarrow \infty$ for $n\rightarrow \infty$? Strengthening previous partial results, we solve this problem affirmatively for all $k>4$, proving that $$f_k(n)=\Omega(n^{1/3}).$$ This leaves only the case $k=4$ open. We also show that a stronger lower bound of order $\sqrt{n}$ holds along an infinite sequence of numbers $n$. Finally, we provide a first non-trivial upper bound on the functions $f_k$ by proving that $$f_k(n)=O\left(\frac{n}{(\log n)^{\Omega(1)}}\right)$$ for every $k\ge 4$. Our proof of the lower bound on $f_k(n)$ involves an intricate analysis of the structure of proper colorings of a modification of an earlier construction due to Jensen, combined with a gluing operation that creates new vertex-critical graphs without small critical edge sets from given such graphs. The upper bound is obtained using a variant of Szemer\'{e}di's regularity lemma due to Conlon and Fox.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

W. H. Organization, World report on vision (2019)

work page 2019

[2] [2]

Zhang, Z

X. Zhang, Z. Xiao, H. Fu, Y . Hu, J. Yuan, Y . Xu, R. Higashita, J. Liu, Attention to region: Region-based integration-and-recalibration networks for nuclear cataract classification using as-oct images, Medical Image Analysis 80 (2022) 102499

work page 2022

[3] [3]

Zhang, Y

X.-Q. Zhang, Y . Hu, Z.-J. Xiao, J.-S. Fang, R. Higashita, J. Liu, Machine learning for cataract classification /grading on ophthalmic imaging modalities: a survey, Machine Intelligence Research 19 (3) (2022) 184–208

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Zhang, Z

X. Zhang, Z. Xiao, B. Yang, X. Wu, R. Higashita, J. Liu, Regional context-based recalibration network for cataract recognition in as-oct, Pattern Recognition 147 (2024) 110069

work page 2024

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W. Wang, J. Zhang, X. Gu, X. Ruan, X. Chen, X. Tan, G. Jin, L. Wang, M. He, N. Congdon, Objective quantification of lens nuclear opacities using swept- source anterior segment optical coherence tomography, British Journal of Oph- thalmology (2021). 26

work page 2021

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Z. Xiao, X. Zhang, B. Zheng, Y . Guo, R. Higashita, J. Liu, Multi-style spatial attention module for cortical cataract classification in as-oct image with super- vised contrastive learning, Computer Methods and Programs in Biomedicine 244 (2024) 107958

work page 2024

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P. Yin, M. Tan, H. Min, Y . Xu, G. Xu, Q. Wu, Y . Tong, H. Risa, J. Liu, Automatic segmentation of cortex and nucleus in anterior segment oct images, in: Compu- tational Pathology and Ophthalmic Medical Image Analysis: First International Workshop, COMPAY 2018, and 5th International Workshop, OMIA 2018, Held in Conjunction with MICCAI 2018, Granada, Spain...

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Zhang, Y

S. Zhang, Y . Yan, P. Yin, Z. Qiu, W. Zhao, G. Cao, W. Chen, J. Yuan, R. Higashita, Q. Wu, M. Tan, J. Liu, Guided m-net for high-resolution biomedical image seg- mentation with weak boundaries, in: Ophthalmic Medical Image Analysis: 6th International Workshop, OMIA 2019, Held in Conjunction with MICCAI 2019, Shenzhen, China, October 17, Proceedings 6, Spr...

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