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REVIEW 2 major objections 5 minor 9 references

Uniform \v{S}olt\'es' hypergraphs and \v{S}olt\'es' weighted graphs

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Uniform Šoltés' hypergraphs exist for every order n≥10, and none have order below 10.

desk verdict Solid lower bound and weighted examples, but the 'every n>=10' theorem has a false equality in Lemma 19 that makes the abundancy construction conditional on a repair. read the letter →

arxiv 2506.07511 v1 pith:OQE3MEIU submitted 2025-06-09 math.CO

Stijn Cambie , Ajay Tiwari This is my paper · ORCID
classification math.CO MSC 05C6505C1205C3505C22
keywords Šoltés'problemWienerindexuniformhypergraphsvertex-transitiveweightedgraphstotaldistancenon-regular
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 hypergraphs whose total pairwise distance, the Wiener index, does not change when any single vertex is deleted (Šoltés' hypergraphs). It establishes that no uniform example has fewer than ten vertices, and that for every $n \ge 10$ a uniform Šoltés' hypergraph of order $n$ exists, with the construction covering most uniformities as well. It also exhibits a non-regular $9$-uniform example on $54$ vertices and infinitely many weighted Šoltés' graphs based on prism graphs with a tuned edge weight. Together with known results for digraphs and signed graphs, this answers the Šoltés' problem positively for the most natural generalizations of graphs.

What carries the argument

The load-bearing construction is a family of vertex-transitive $k$-uniform hypergraphs indexed by integers $s,t$ with $n = \binom{s}{2} - t$, $k = n - (t+2s+1)$, and hyperedges $e_i = \{i, i+2s+k-1\} \cup \{i+s+1, \ldots, i+s+k-2\}$ (indices modulo $n$). These hyperedges are chosen so that $H$ has diameter $1$ while $H\setminus v$ has diameter at most $2$, and Lemma 19's count of exactly $n-1$ non-adjacent pairs in $H\setminus v$ turns the identity $\binom{n-1}{2} + (n-1) = \binom{n}{2}$ into the Šoltés' equality $W(H\setminus v) = W(H)$. For the weighted graphs, the mechanism is the vertex-transitive prism $C_{2k} \,\square\, K_2$ with cycle edges of weight $1$ and the remaining edges of weight $x$, chosen so that for each vertex the sum of deletion detours equals that vertex's transmission.

What would settle it

Recompute the pair count in Lemma 19 for a concrete parameter choice, such as $s=15$, $t=0$, $n=105$, $k=74$; if the number of non-adjacent pairs in $H\setminus v$ is not $104$, the abundance construction fails for that parameter. The displayed block-covering equality in the lemma's proof can be checked directly for these numbers.

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Extended reading notes

Core claim

The central discovery is that the Šoltés' phenomenon extends to uniform hypergraphs with a sharp order threshold: no uniform Šoltés' hypergraph has order below $10$, and for every $n \ge 10$ there is a vertex-transitive, $k$-uniform construction satisfying $W(H \setminus v) = W(H) = \binom{n}{2}$. The construction sets $n = \binom{s}{2} - t$ and $k = n - (t+2s+1)$, with hyperedges $e_i = \{i, i+2s+k-1\} \cup \{i+s+1, \dots, i+s+k-2\}$ (indices modulo $n$), and the proof rests on counting exactly $n-1$ pairs of vertices at distance two in $H\setminus v$. In addition, a $9$-uniform hypergraph on $54$ vertices with two vertex orbits is a Šoltés' hypergraph but is not regular, and an infinite family of weighted prism graphs $C_{2k} \,\square\, K_2$ with edge weights $1$ and $x = \frac{2k^2-6k+16}{k^2-9k+12}$ satisfies $W(G) = W(G\setminus v)$ for every vertex $v$.

Load-bearing premise

The statement that uniform Šoltés' hypergraphs exist for every order $n \ge 10$ depends on Lemma 19's exact count: after deleting one vertex from the constructed hypergraph, exactly $n-1$ pairs of vertices remain at distance two, and if that count is wrong for some parameter choice the construction stops working and only the computer-found orders up to $100$ remain.

Editorial extensions

If this is right

  • For every order $n \ge 10$ there is at least one uniform Šoltés' hypergraph, so the hypergraph analogue of Šoltés' problem has no order obstruction beyond the threshold 10.
  • The $9$-uniform example on $54$ vertices shows that regularity is not necessary for a hypergraph to be Šoltés'.
  • The weighted prism family gives infinitely many weighted Šoltés' graphs, and rescaling the weights yields examples with positive integer weights and greatest common divisor 1.
  • Together with earlier results for digraphs and signed graphs, the paper concludes that the most natural generalizations of graphs all admit infinitely many Šoltés' examples.
  • Since the main construction covers 75% of uniformities and the computer searches cover $4 \le k \le 10^7$ except for eight uniformities, the remaining open uniformity question is essentially confined to small cases.

Reading between the lines

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

  • Editorial inference: the same exact-count technique in Lemma 19 could be adapted to the $r$-parameter generalization described in Remark 20, which might turn the 75% uniformity coverage into a proof for every uniformity $k \ge 4$ once the analogous pair count is verified.
  • Editorial inference: the weighted-prism construction suggests a general recipe—start from a vertex-transitive graph and tune one weight so that the deletion detour sum cancels the transmission—which could produce weighted Šoltés' graphs with more than two weight values or on other product families.
  • Editorial inference: the remaining computer checks in the order-10 threshold proof are small enough that they could likely be replaced by an analytic double-counting argument, removing the computational component from the lower-bound theorem.
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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

2 major / 5 minor

Summary. The paper studies Šoltés' problem in two generalisations: uniform hypergraphs and weighted graphs. It claims that every uniform Šoltés' hypergraph has order at least 10, that for every n ≥ 10 there exists a uniform Šoltés' hypergraph of order n, that there exists a non-regular uniform example, and that there are infinitely many weighted Šoltés' graphs. Section 2 gives a computer-assisted exclusion of all orders up to 9, Section 3 constructs vertex-transitive k-uniform hypergraphs for all n ≥ 100 and cites computer examples for 10 ≤ n ≤ 100, with Lemma 19 as the key exact-count step, and Section 4 gives an irregular 9-uniform example. The introduction also contains an explicit weighted prism construction.

Significance. The questions addressed are natural, and positive abundancy results for hypergraphs and weighted graphs would continue a line of work on digraphs and signed graphs. The lower-bound analysis in Section 2 is substantial, combining structural lemmas with finite checks, and the weighted prism example is explicit and algebraically verifiable. The paper's computational component is a strength, but the central 'every n ≥ 10' claim rests on Lemma 19, whose proof currently contains a concrete numerical error. The abstract also asserts a uniformity-abundancy statement that is not proven in the text. If the Section 3 gap can be repaired, the paper would be a solid contribution; as written, the main existence theorem is conditional.

major comments (2)
  1. [Section 3, Lemma 19] The displayed covering equality in the proof of Lemma 19 is numerically false for the stated parameters. For s = 15, t = 0, n = 105, k = 74, the asserted identity ⌈t/2⌉ + s + k − 2 = n − (⌈t/2⌉ + s + 1) reads 87 = 89. The interval description of e_{⌈t/2⌉}, the covering claims for e_{−s} and e_{t+s+2}, and the subsequent assertion that non-neighbours are confined to [1, s] × [n−s, n−1] all depend on this equality. Since the exact count of n − 1 non-adjacent pairs in H\v is the only step that yields W(H\v) = C(n−1,2) + (n−1) = C(n,2), the proof of 'there exists a uniform Šoltés' hypergraph of every order n ≥ 10' is not established for n ≥ 100. The finite examples cover only 10 ≤ n ≤ 100. This needs either corrected parameters/endpoints with a complete proof of the pair count, or a machine-checkable certificate for the count for all n ≥ 100.
  2. [Abstract and Section 3, Remark 20] The abstract and introduction claim existence of uniform Šoltés' hypergraphs for 'almost every order or uniformity' and state as a main result that 'for most integers k ≥ 4, there exists a k-uniform Šoltés' hypergraph'. However, Remark 20 does not prove this: it says the construction provides only 75% of uniformities, lists eight missing values up to 10^7, and explicitly states that 'one can expect' all uniformities to occur and that other cases were not worked out. This is a conjecture or empirical observation, not a theorem. The claim should be removed from the list of proved results or clearly labelled as open/conjectural.
minor comments (5)
  1. [Section 2, Theorem 7] In the proof of Theorem 7, the first sentence reads 'if H is 3-uniform Šoltés' hypergraph with order 8', but the theorem concerns order 9; this is a typo.
  2. [Section 2, Proposition 5] The phrase 'the remaining case is n = 9 and u = 5' should read 'k = 5', not 'u = 5'.
  3. [Figure 1 caption] The caption contains the typo 'weigthed'; it should be 'weighted'.
  4. [References and reproducibility] The GitHub repository linked in [4] is not pinned to a commit or version, so the finite computer checks cannot be reproduced exactly from the manuscript; adding a commit hash would improve reproducibility.
  5. [Section 3, Lemma 19] The final summation in Lemma 19 runs only to s−1 for the second range, while the preceding sentence describes non-neighbours for t+2 ≤ j ≤ s. Since s − s = 0, the omission is harmless, but the text should make this explicit to avoid confusion.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central claims rest on explicit in-paper constructions (Lemma 19, Example 3, Example 21), with no fitted parameter renamed as a prediction; residual self-citations cover small finite ranges and code-verifiable examples, while Lemma 19's false interval equality is a correctness gap, not a circular reduction.

full rationale

The central derivations are self-contained constructions rather than fits repackaged as predictions. Section 3's abundancy claim rests on Lemma 19, which computes the number of non-adjacent pairs in H\v as n−1; if that count is correct it yields W(H\v)=C(n−1,2)+(n−1)=C(n,2)=W(H) with no parameter fitted to the target equation, since the hypergraph is an explicit function of n, s, and t and the count is a verification. Example 3 similarly solves x from W(G)=W(G\v) transparently ('by the choice of x'), which is legitimate for an existence claim rather than a hidden prediction. The residual self-reliance is minor: [3] supplies the n≤5 exclusion and the question list, and [9] supplies the explicitly constructed examples for 10≤n≤100, verified by the authors' code [4]; these are small finite ranges and code-reproduced evidence, and the central constructions are stated in full in the paper, so no load-bearing step reduces to an unverified self-citation or an imported uniqueness theorem. Two non-circular risks should be weighed: Lemma 19's displayed equality ⌈t/2⌉+s+k−2 = n−(⌈t/2⌉+s+1) is numerically false (for s=15, t=0, n=105, k=74 it reads 87=89, and in general 2⌈t/2⌉−t=2 never holds), so the exact count of n−1 non-adjacent pairs is not established as written and the n≥100 abundancy theorem is conditional on repairing that interval arithmetic; and the GitHub verifications [4] are cited without a commit hash, so the computational checks are not pinned to a specific code version. Neither gap makes any derivation equivalent to its inputs, so the circularity score stays low.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The analytic core uses three external theorems: [3, Thm 3] (no uniform Šoltés' hypergraph below order 6), [5, Thm 1] (tight path maximizes Wiener index of uniform hypergraphs), and [7, Thm 1] (colex plus clique maximizes graph Wiener index given order and size). Above that, the paper introduces a family of finite computer verifications (small-order exclusions, remaining cases for (8,3) and (9,3), the irregular example, the uniformity census up to 10^7) that are referenced to a GitHub repo but not shown in the text, and the k=2 exclusion rests on an unpublished check by Stevanovic. The Section 3 construction is credited to co-author Tiwari's thesis [9]. Free parameters: the weight x in Example 3 (solved from the target equality) and the searched edge pattern of Example 21.

free parameters (3)
  • weight x of cross edges in the prism construction (Example 3) = x = (2k^2 - 6k + 16)/(k^2 - 9k + 12), for k >= 20
    The weight is solved from the target equality W(G) = W(G-v): it is exactly the value making the detour sum equal the vertex transmission. Legitimate for an existence construction, but it is a hand-chosen value, not derived independently.
  • edge pattern of the irregular example (Example 21) = 27 hyperedges {a, a+1, a+2, a+3, a+4, a+5, a+7, a+16, a+18} mod 54, a even
    The specific steps (five +1s, then +2, +9, +2) and the 54-vertex, 9-uniform parameters were found by search and verified by computation; the text gives no derivation of why this pattern has W(H) = 2349.
  • construction parameters s, t (Section 3)
    Existence parameters: write n = C(s,2) - t with 0 <= t <= s-2 and set k = n - (t + 2s + 1). Not fitted to data, but the claimed pair count and the 'for every n >= 100' theorem depend on these ranges.
assumptions (6)
  • standard math [3, Thm 3]: there is no uniform Soltes' hypergraph of order at most 5
    Invoked in Proposition 5 to start the exclusion at n = 6. Published in Cambie 2024; shared first author with this paper.
  • standard math [5, Thm 1]: the k-uniform hypergraph of maximum Wiener index for fixed order and uniformity is the tight path; P^3_7 and P^3_8 have Wiener index 38 and 57
    Used in Proposition 5 and Theorems 6 and 7 to bound W(H-v) from above.
  • standard math [7, Thm 1]: for graphs, the maximum Wiener index given order and size is attained by a colex graph concatenated with a clique, giving bounds like W(G) <= 42 - m for order 7
    Used in Theorems 6 and 7 to bound the Wiener index of the underlying graph of H-v.
  • domain assumption C11 is the only Soltes' graph of order at most 12 (unpublished computer check by D. Stevanovic)
    This is the sole basis for excluding k = 2 in Proposition 5; no published reference is provided.
  • domain assumption Finite computer verifications: no 3-uniform example on 7 vertices with m <= 10 edges; no 4-uniform order-8 with m <= 6; no 5-uniform order-9 with m <= 6; W(H-v) <= 19 and <= 34 bounds; remaining (8,3) and (9,3) cases excluded by computer
    Asserted in Proposition 5 and Theorems 6 and 7 with pointers to the GitHub repository [4]; the computations are not shown in the text.
  • domain assumption Uniformity census of Remark 20: the r = 1 construction covers 75% of uniformities, and for 4 <= k <= 10^7 only 8 uniformities lack examples: 905, 1301, 1721, 2801, 9641, 11969, 52109, 5335709
    These coverage claims are computational and not demonstrated in the text; they support the abstract's 'almost every uniformity' phrasing.

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Pith. "Pith review of Uniform \v{S}olt\'es' hypergraphs and \v{S}olt\'es' weighted graphs." pith.science (2026). https://pith.science/paper/OQE3MEIU

@misc{pith2026250607511,
  author       = {Pith},
  title        = {Pith review of: Uniform \vSolt\'es' hypergraphs and \vSolt\'es' weighted graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQE3MEIU}},
  note         = {Machine review of arXiv:2506.07511}
}
abstract

A \v{S}olt\'es' hypergraph is a hypergraph for which the removal of any of its vertices does not change its total distance. We prove that every uniform \v{S}olt\'es' hypergraph has order at least $10$, there exist uniform \v{S}olt\'es' hypergraphs for almost every order or uniformity, and there exist a non-regular uniform \v{S}olt\'es' hypergraph. By also providing infinitely many weighted \v{S}olt\'es' graphs, we conclude that \v{S}olt\'es' problem can be answered positively for the most natural generalisations of graphs.

Figures

Figures reproduced from arXiv: 2506.07511 by the authors.

Figure 1
Figure 1. A weigthed Šoltés’ graph of order 80 Overview and outline of the paper In Section 2, we prove that there is no uniform Šoltés’ hypergraph with order bounded by 9. We do so by applying different strategies for different pairs (n, k) of order and uniformity on a possible Šoltés’ hypergraph H. • The case k = 2 was excluded by a brute force computer check. • If k ≥ n − 3 ≥ 3, the connected hypergraph H \ v of order n − … view at source ↗

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    S. Cambie. Abundancy ofz-Šoltés’ digraphs.arXiv e-prints, page arXiv:2501.00102, Dec. 2024

    work page Pith review arXiv 2024

  2. [2]

    S. Cambie. Towards the essence of Šoltés’ problem.arXiv e-prints, page arXiv:2406.03451, June 2024

    work page Pith review arXiv 2024

  3. [3]

    S. Cambie. Šoltés’ hypergraphs.arXiv e-prints, page arXiv:2406.01504, June 2024

    work page Pith review arXiv 2024

  4. [4]

    uniform šoltés’ hypergraphs and šoltés’ weighted graphs

    S. Cambie. Code and data for the paper “uniform šoltés’ hypergraphs and šoltés’ weighted graphs”. https://github.com/StijnCambie/Soltes/tree/main/UnifSoltesHypergraphs, 2025

    work page 2025

  5. [5]

    The maximum Wiener index of a uniform hypergraph

    S. Cambie, E. Győri, N. Salia, C. Tompkins, and J. Tuite. The maximum Wiener index of a uniform hypergraph.arXiv e-prints, page arXiv:2302.08686, Feb. 2023. 9

    work page Pith review arXiv 2023

  6. [6]

    M. Knor, R. Škrekovski, and A. Tepeh. Selected topics on Wiener index.Ars Math. Contemp., 24(4):31, 2024. Id/No 7

    work page 2024

  7. [7]

    L. Šoltés. Transmission in graphs: a bound and vertex removing.Mathematica Slovaca, 41(1):11–16, 1991

    work page 1991

  8. [8]

    S. Spiro. The Wiener index of signed graphs.Appl. Math. Comput., 416:10, 2022. Id/No 126755

    work page 2022

Show all 9 references
  1. [9]

    A. Tiwari. Uniform Šoltés’ Hypergraphs.Master thesis, 2025. 10

    2025

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