Infinitely many counterexamples to a conjecture of Lov\'asz

Aida Abiad; Christoph Spiegel; Frederik Garbe; Xavier Povill

REVIEW 2 major objections 5 minor 31 references

Infinitely many counterexamples to a conjecture of Lov\'asz

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that the line hypergraphs of the generalized Petersen graphs GP(5k+11,2), for every k≥0, are counterexamples to Lovász's 1975 conjecture on reducing the matching number by deleting r−1 vertices, and it provides the first…

desk verdict The r=3 infinite family is real, the induction is sound, and the only real caveat is the computationally asserted base cases, which are a small gap rather than a fatal one. read the letter →

arxiv 2506.21286 v2 pith:4GXJQ36Z submitted 2025-06-26 math.CO

Aida Abiad , Frederik Garbe , Xavier Povill , Christoph Spiegel This is my paper · ORCID

classification math.CO MSC 05C6505C6905C70

keywords LovászconjectureRyserhypergraphmatchingnumberlinegeneralizedPetersengraphsindependencer-partitehypergraphscounterexample

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

Lovász's conjecture from 1975 says that in any r-partite r-uniform hypergraph one can remove r−1 vertices and thereby shrink the matching number. This paper proves the conjecture false for r=3 in a strong sense: the line hypergraphs of the generalized Petersen graphs GP(5k+11,2), for every k≥0, are counterexamples. The smallest such example comes from a cubic graph on only 22 vertices, far smaller than the two sporadic counterexamples previously known. The paper also provides the first counterexamples for r=4, constructed as line hypergraphs of certain 4-regular Cayley-generalized Petersen graphs. If correct, this shows Lovász's proposed path toward Ryser's conjecture has infinitely many blocking examples at uniformity 3 and at least isolated failures at uniformity 4.

What carries the argument

The line hypergraph L(G) has as vertices the edges of G and as hyperedges the incidence sets S(v); it is r-uniform and r-partite exactly when G is r-regular and r-edge-colorable, and independent sets of G correspond bijectively to matchings of L(G). Lemma 7 converts Lovász's deletion condition into the graph statement α(G−⋃_{e∈S}e)=α(G) for every set S of r−1 edges. Theorem 13 verifies this statement for GP(5k+11,2) by induction, using the known formula α(GP(n,2))=⌊4n/5⌋, the bound that five consecutive rungs contain at most four independent vertices, and the existence of Tait colorings. The induction removes five consecutive rungs, applies the smaller case, and then reassembles a maximum independent set in the larger graph that avoids the two removed edges.

What would settle it

Run an independent computation of α(GP(5k+11,2)−{u,v,w,x}) for all pairs of edges uv,wx when k=0,1,2; if any result is smaller than α(GP(5k+11,2)), then Theorem 13 fails and with it the proof of the infinite family.

Watch

Extended reading notes

Core claim

The paper proves Theorem 2: for every k≥0, the line hypergraph of GP(5k+11,2) is a 3-uniform, 3-partite hypergraph whose matching number cannot be decreased by deleting any two vertices. The core identity is α(GP(5k+11,2)−{u,v,w,x})=α(GP(5k+11,2)) for any two edges uv,wx. Because the graph is 3-edge-colorable, this identity makes the line hypergraph a counterexample to Lovász's conjecture. The paper also proves Theorem 3: several 4-regular graphs, including CGP(2,Z35;5,7;15,0) on 70 vertices, give the first counterexamples for r=4.

Load-bearing premise

The infinite family proof is an induction whose three base cases k=0,1,2 are asserted to be 'easily verified computationally' but are not shown in the paper; if any of those three checks is wrong, the induction has no valid base and Theorem 2 collapses.

Editorial extensions

If this is right

For every k≥0, the matching number of the line hypergraph of GP(5k+11,2) stays unchanged after any two vertices are removed, so Lovász's conjecture fails infinitely often for r=3.
The smallest member of this infinite family is built from a 22-vertex cubic graph, giving an explicit counterexample much smaller than the prior 102-vertex example.
The r=4 counterexamples, starting with a 70-vertex quartic graph, show that the failure is not confined to uniformity 3.
None of the constructed hypergraphs contradict Ryser's conjecture or are extremal for it, so Ryser's conjecture can still hold even though this route to it is blocked.
Some of the counterexamples are Cayley graphs and some are not, so edge-transitivity and the Cayley property are not essential to the phenomenon.

Reading between the lines

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

Editorial extension: the gluing construction in Theorem 13 is not obviously tied to the parameter 2 in GP(n,2), so similar infinite families may exist among other generalized Petersen graphs; a computer search over GP(n,k) with n≡1 mod 5 would test this directly.
Editorial extension: because the r=4 examples are found by a finite computer search over Cayley-generalized Petersen graphs, and the paper leaves open whether an infinite r=4 family exists, the natural next question is whether such an infinite family exists; searching permutation graphs with additional structural assumptions would be a first place to look.
Editorial extension: the weakened version of Lovász's conjecture, which allows deleting k(r−1) vertices to decrease the matching number by at least k, is verified by the authors for small r=3 counterexamples but not for r=4; checking it for the 70-vertex quartic example is a concrete open computation.
Editorial extension: all counterexamples here are line hypergraphs, so the failure of Lovász's conjecture might be specific to this construction; testing non-line-hypergraph r-partite hypergraphs for analogous failures would clarify how broadly the conjecture fails.

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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 one thing to know: the infinite family for r=3 is real. The proof is an induction on k, and the inductive step is sound; I checked the gluing and the cross-edge cases. The only place I had to take anything on faith is the computationally asserted base cases k=0,1,2, and that is a small gap, not a fatal one.

What's new: Clow, Haxell, and Mohar had two sporadic 3-uniform counterexamples. This paper shows all generalized Petersen graphs GP(5k+11,2) have line hypergraphs violating Lovász's conjecture, starting at order 22. That answers their Question 4.1 in the strongest sense. It also produces the first r=4 examples, via a computer search over Cayley-generalized Petersen graphs, and a 36-vertex cubic Cayley graph, which addresses the "Cayley" question. The tables in Section 4 give a nice census.

The main trick is solid. Lemma 12 is a short pigeonhole argument that after deleting any two edges you still have a 10-rung block untouched; Lemma 10 bounds the independent set on five consecutive rungs; the induction in Theorem 13 then shifts the independent set across the gluing. I don't see a hole in the cross-segment edges listed at the end of Theorem 13. Using the known independence formula for GP(n,2) is legitimate.

The soft spots are minor but worth noting. The base cases k=0,1,2 for the induction are described only as 'easily verified computationally,' with code linked on GitHub. Since the inductive step is stated for k≥2, the k=2 check is genuinely load-bearing: if it were wrong, the whole induction falls. The code is not pinned to a commit, so a referee can't easily re-run the exact verification. That's easily fixed: include an explicit certificate or a short printed verification in the appendix. Same for the r=4 table: everything is computational, and the search is not claimed to be exhaustive, but the reader should be able to reproduce the checks. I'd like to see a reproducibility note with library versions and a checksum.

One point where I disagree with the reader's take: the abstract doesn't overstate the r≥4 answer. It says 'specific counterexamples for r=4,' which is exactly what they deliver. The open question of infinite families for r≥4 remains open, but that's not misrepresented.

Bottom line: this is a strong, clean contribution to a 50-year-old conjecture. The proof is convincing, the citation to prior work is accurate, and the computational parts are small and verifiable. Send it to a referee; with a request to make the base-case verification reproducible, it should be accepted.

Referee Report

2 major / 5 minor

Summary. The paper constructs an infinite family of cubic graphs, the generalized Petersen graphs GP(5k+11,2) for k≥0, whose line hypergraphs are counterexamples to Lovász's conjecture for r=3, thereby answering an open question of Clow, Haxell, and Mohar. The proof centers on Theorem 13, which asserts that for any two edges of GP(5k+11,2), removing their endpoints does not change the independence number; together with a Tait coloring and Lemma 7, this implies the line hypergraph is a counterexample. The inductive step of Theorem 13 is proven rigorously for k≥2, while the base cases k=0,1,2 are only asserted to be computationally verified. The paper also presents the first counterexamples for r=4, obtained computationally from Cayley-generalized Petersen graphs, with several specific graphs listed and made available via graph6 strings and code.

Significance. If the computational verifications are correct, this is a substantial advance: it provides the first infinite family of counterexamples to Lovász's conjecture for r=3, greatly improving on the two sporadic examples of Clow et al., and it gives the first counterexamples for r=4. The induction argument in Theorem 13 is elegant and uses classical results on generalized Petersen graphs in a clean, parameter-free way. The paper is commendable for making its computational parts reproducible in principle: the authors provide a GitHub repository with code and graph descriptions. However, the lack of in-paper details or certificates for the load-bearing computational steps prevents the current version from being fully self-contained.

major comments (2)

[Section 3, proof of Theorem 13] The base cases k=0,1,2 are asserted to be 'easily verified computationally' with only a GitHub link, but since the inductive step is proved only for k≥2, the k=2 case is essential for Theorem 2, and k=0 and k=1 are needed for the full statement 'every k≥0' and for the smallest-order example GP(11,2). This is a load-bearing computational premise rather than a minor remark; please provide a complete verification, such as a table of all edge pairs with the resulting independence numbers, or include the verification script and its output in the paper, and pin the repository to a specific commit.
[Section 4, Theorem 3] The quartic counterexamples are established entirely by computational verification, but the paper only states 'The properties of all of the graphs have been verified computationally' and points to a GitHub repository; it does not specify the exact condition checked, the number of 3-edge subsets tested for each graph, or provide any output or certificate for the listed examples. Since Theorem 3 rests on these checks, please document the verification procedure and provide the relevant code and results so that a reader can reproduce the computation and confirm the claims.

minor comments (5)

[Section 3, first paragraph] The phrase 'Out infinite family' should be 'Our infinite family'.
[Definition 8] The formula defining S_{i,ℓ} is garbled in the text ('Si,ℓ := Si+ℓ−1 i Rj'); it should be the union of ℓ consecutive rungs, e.g., S_{i,ℓ} = ⋃_{j=i}^{i+ℓ-1} R_j.
[Proof of Lemma 12] The sentence 'we will imply a natural order 1 < 2 < ... < n on the rungs' should read 'we will impose a natural order'.
[Abstract] The abstract says the counterexample of Clow, Haxell, and Mohar uses the 'line graph' of a 3-regular graph, but it should say 'line hypergraph' to be consistent with the rest of the paper.
[Section 4 and GitHub link] For reproducibility, the GitHub repository should be archived (for example, via Zenodo) or at least a commit hash should be provided, so that the code used for the computational verifications is permanently available in the exact version used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the infinite-family proof is an induction using external independence and Tait-coloring results, with no fitted parameter or prediction reducing to an input; the computationally asserted base cases are a reproducibility gap, not a circular step.

full rationale

The derivation chain is self-contained. Lemma 7 is restated from Clow, Haxell, and Mohar [8, Lemma 2.2], an external characterization of Lovasz counterexamples in terms of independence numbers; the paper's use of it does not assume the target result. The independence numbers of GP(n,2) and the Tait-coloring result are cited from [14] and [6], both independent published results. The proof of Theorem 13 is a genuine induction: for k >= 2 it applies Lemma 12, constructs \tilde G = GP(5k+11,2), invokes the induction hypothesis, and builds an independent set of the required size; the argument is not defined in terms of its conclusion. The only asserted inputs are the base cases k=0,1,2, which are stated to be 'easily verified computationally' with code linked on GitHub; this is an omitted verification rather than a circular reduction, because no fitted parameter is being relabeled as a prediction and no self-citation is load-bearing. Similarly, the r=4 examples in Section 4 are found by computational search and verified with external tools (nauty, cliquer, SageMath), not by fitting a model to the very data being predicted. No equation in the paper reduces to a previous equation by construction, and no uniqueness theorem from the authors' prior work is invoked. The appropriate finding is no significant circularity.

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

The proof relies on three cited structural results about generalized Petersen graphs and on a computational check of three base cases. No free parameters are fitted. The new CGP family for r=4 is a definition, not an invented entity with independent physical evidence.

assumptions (4)

domain assumption For n >= 5, alpha(GP(n,2)) = floor(4n/5) (Fox, Gera, Stanica, Theorem 2.5 in [14]).
Used in the proof of Theorem 13 to equate the target size 4k+12 with alpha(GP(5k+11,2)); the paper does not reprove this formula.
domain assumption For n >= 5, every segment of 5 consecutive rungs in GP(n,2) has independence number at most 4 ([14, Lemma 2.4]).
Used in the induction step of Theorem 13 to assert that the shifted independent set misses at least one of the five rungs R_{n-9}..R_{n-5}.
domain assumption Every generalized Petersen graph except GP(5,2) has a Tait 3-edge-coloring (Castagna and Prins [6]).
With n=5k+11 never equal to 5, this makes GP(5k+11,2) 3-edge-colorable, so its line hypergraph is 3-partite by Remark 5, a necessary condition for the Lovasz conjecture setting.
ad hoc to paper The base cases k=0,1,2 in Theorem 13 satisfy the independence-number condition, as verified by the linked computer code.
The induction in Theorem 13 requires these three finite cases, and the paper provides no written proof for them, only the GitHub link; the correctness of the whole infinite family depends on this computation.

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Pith. "Pith review of Infinitely many counterexamples to a conjecture of Lov\'asz." pith.science (2026). https://pith.science/paper/4GXJQ36Z

@misc{pith2026250621286,
  author       = {Pith},
  title        = {Pith review of: Infinitely many counterexamples to a conjecture of Lov\'asz},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4GXJQ36Z}},
  note         = {Machine review of arXiv:2506.21286}
}

abstract

Motivated by the well-known conjecture of Ryser which relates maximum matchings to minimum vertex covers in $r$-partite $r$-uniform hypergraphs, Lov\'asz formulated a stronger conjecture. It states that one can always reduce the matching number by removing $r-1$ vertices. This conjecture was very recently disproven for $r=3$ by Clow, Haxell, and Mohar using the line graph of a $3$-regular graph of order $102$. Building on this, we describe a simple infinite family of counterexamples based on generalized Petersen graphs for the case $r=3$ and give specific counterexamples for $r=4$.

Figures

Figures reproduced from arXiv: 2506.21286 by the authors.

**Figure 1.** The 3-regular generalized Petersen graph GP(11, 2) on 22 vertices whose line hypergraph is a counterexample for Lov´asz’ conjecture for r = 3. It generalizes to the infinite family of graphs GP(5k + 11, 2) whose line hypergraphs are counterexamples to Lov´asz’ conjecture for r = 3 and any k ≥ 0. The case r = 2 also follows from K˝onig’s theorem, since removing any vertex from a minimum vertex cover creates a graph w… view at source ↗

**Figure 2.** The 4-regular graph CGP(2, Z35; 5, 7; 15, 0) on 70 vertices whose line hypergraph is a counterexample to Lov´asz’ conjecture for r = 4. Theorem 2. There exists an infinite family of cubic graphs whose line hypergraphs are counterexamples to Lov´asz’ conjecture for r = 3. We also give a first answer to the second question by providing several specific counterexamples for r = 4, likewise derived as the line hypergrap… view at source ↗

**Figure 3.** A 3-regular Cayley graph in Z36 whose line hypergraph is a counterexample to Lov´asz’ conjecture for r = 3. 2 Preliminaries Let us denote the vertex and edge set of a graph or hypergraph G respectively by VG and EG. Given a graph G, we let G[S] denote the graph induced by some S ⊆ VG, α(G) the independence number, and S(v) = {e ∈ EG | v ∈ E} the incidence set of some vertex v ∈ VG. Definition 4. The line hypergraph … view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Illustration for the proof of Theorem 13. Note that by construction of the independent set I no vertex from Ri1 or Ri1+5 is selected for I. Hence the four marked vertices defined by (iv) are not adjacent to any vertex in I. bi1+4bi1+6 are contained in I, see also [PIT…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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