Pancyclicity in hypergraphs with large uniformity

Isaiah Hollars; Ruth Luo; Teegan Bailey; Yupei Li

REVIEW 3 major objections 5 minor 24 references

Pancyclicity in hypergraphs with large uniformity

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Sharp minimum-degree conditions force Berge cycles of every length in large-uniformity hypergraphs.

desk verdict A genuine completion of the large-r pancyclicity threshold, with sound but intricate proofs; referee should focus on Theorem 9 Case 4. read the letter →

arxiv 2505.00130 v1 pith:6BFUK6CO submitted 2025-04-30 math.CO

Teegan Bailey , Isaiah Hollars , Yupei Li , Ruth Luo This is my paper · ORCID

classification math.CO MSC 05C6505C3505C38

keywords hypergraphBergecyclespancyclicminimumdegreeextremalcombinatoricsHamiltonianr-uniformhypergraphsbipancyclic

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 proves exact minimum-degree thresholds that force an n-vertex r-uniform hypergraph to contain Berge cycles of every length from 2 to n, for the large-uniformity range r ≥ floor((n-1)/2)-1 and sufficiently large n. The thresholds are the same ones that already force a Hamiltonian Berge cycle, so the result is a hypergraph analogue of the classical graph meta-conjecture that Hamiltonian strength should imply pancyclicity. Below r = n/2 the required minimum degree is binomial(floor((n-1)/2), r-1)+1; at or above r = n/2 it is simply r. Three extremal constructions show that lowering the minimum degree by one can destroy the Hamiltonian cycle, so the conditions are sharp. Together with earlier work for smaller r, this closes the pancyclicity question for all uniformities when n is large.

What carries the argument

The shifting function S_s maps vertex indices forward by s with a single fold, and Lemma 11 shows that if S_s(U_0) meets the Hamiltonian edge e_0, then H contains an (n-s+1)-Berge cycle; contrapositively, a missing cycle forces the disjointness condition S_s(U_0) ∩ e_0 = ∅. The k-self-shift-complementary (k-SSC) sets — subsets A of the n-cycle with |A| = n/2 and A ∩ (k+A) = ∅ — classify the vertex footprints that a large extra edge can have when no k-chord exists, and the SSC-structure proposition forces such an A to alternate inside cosets of a subgroup. In the borderline-uniformity case, a cycle-compatible graph is constructed whose edges correspond to pairs with large co-degree in the extra edges, and known weak-pancyclicity and weak-bipancyclicity results for graphs are applied to force cycles of every length in that graph, which then lift to Berge cycles.

What would settle it

Search the boundary cases, for instance n = 31 with r = 14, for a 14-uniform hypergraph with minimum degree at least 106 that is missing a Berge cycle of some specific length; finding one would refute Theorem 5, while a proof that every such hypergraph has all lengths, and that the only degree-105 examples fail, would confirm the claimed sharpness.

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Extended reading notes

Core claim

The central claim, Theorem 5, is that for r ≥ floor((n-1)/2)-1, every n-vertex r-uniform hypergraph meeting the stated minimum-degree condition contains Berge cycles of all lengths 2 through n. The proof begins with the Hamiltonian Berge cycle C guaranteed by the same degree condition, then studies the edges outside C. The key observation is that if such an extra edge contains a k-chord, the hypergraph immediately has a (k+1)-Berge cycle. The bulk of the argument shows that if some length were missing, the footprints of the extra edges would have to form a rigid alternating pattern, a k-self-shift-complementary set, whose structure contradicts the degree condition. For the borderline case r = floor((n-1)/2)-1, a cycle-compatible auxiliary graph is built and shown to be weakly pancyclic, so cycles in the graph lift to Berge cycles of the same length. The three boundary constructions fail to be Hamiltonian and have minimum degree exactly one below the thresholds, establishing sharpness.

Load-bearing premise

The proof uses, without re-proving it, an external theorem saying that the same minimum-degree conditions already force a Hamiltonian Berge cycle; if that theorem had a flaw or different thresholds, Theorem 5 would not follow.

Editorial extensions

If this is right

The large-uniformity range r ≥ floor((n-1)/2)-1 now has sharp minimum-degree conditions for pancyclicity, completing the picture together with the earlier small-uniformity result.
All three boundary constructions have minimum degree exactly one below the theorem's thresholds and fail even to be Hamiltonian, so no uniform weakening of the degree condition can force pancyclicity.
For r ≥ n/2, a single vertex lying in c_r extra edges beyond the Hamiltonian cycle suffices, with c_r = 1 when r > n/2 and c_r = 6 when r = floor((n-1)/2).
In the borderline case r = floor((n-1)/2)-1, the numerical condition n ≥ 31 can be relaxed to a structural one: each vertex in at least 5(r-1)+2 extra edges forces pancyclicity.

Reading between the lines

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

The numerical thresholds n ≥ 31 and n ≥ 55 are probably not optimal; the proof leaves slack in the inequalities, so the first genuinely hard examples may occur at smaller n than the theorem assumes.
The paper's Question 2 — whether every sufficiently large Hamiltonian r-uniform hypergraph is pancyclic — is answered positively for r > n/2 and negatively when r divides n and n ≥ r^2+3r; the unresolved interval near r = sqrt(n(1-o(1))) is a natural next target for the cycle-compatibility method.
If the six-extra-edge condition in Theorem 9 could be lowered to one, the r = floor((n-1)/2) case would reduce to the authors' Question 1, which would give a very tight bond between Hamiltonicity and pancyclicity.

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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

Thanks for the report. I read the paper and think the reader's ACCEPT is right, with the same moderate confidence.

The new content is real: Theorem 5 completes the hypergraph pancyclicity story for r >= ~n/2 by proving exact thresholds in the previously open range. Theorems 9 and 10 are the engine: they show that a Hamiltonian Berge cycle plus enough extra edges forces all cycle lengths. That is a new structural mechanism, not just a tweak of the Hamiltonian proof.

I went through the reductions and they hold. The incidence-graph argument in Section 2 is elegant and correct. The shifting-function lemma is clean. Theorem 10's cycle-compatibility setup (Claim 15) is a nice idea, and the counting in Claim 14 checks out. Theorem 9 is longer, but the cases are organized and the use of SSC sets is legitimate.

Soft spots: The proof leans on Theorem 3 from Kostochka–Luo–McCourt without re-proving it. That's fine in practice—it's a published theorem—but it means the paper's validity is tied to that external result. The hardest part is Theorem 9 Case 4 (n=2r+2). Claims 12 and 13 are plausible, and the final interval/SSC arguments are intricate. I didn't find a hole, but this is exactly where I'd want a second pair of eyes. The paper's own n≥31 and n≥55 thresholds are not optimal, and the authors admit this; that doesn't affect the theorem. One small blemish: in the concluding remarks, they cite [11] as unavailable, so that part of the discussion is unverified.

Bottom line: This is a solid contribution, worth a serious referee. It should be published in a good combinatorics journal after a careful check of Case 4. I'll cite it if I write anything on Berge cycles.

Referee Report

3 major / 5 minor

Summary. The paper proves sharp minimum degree conditions for Berge pancyclicity in n-vertex r-uniform hypergraphs when r ≥ floor((n-1)/2)-1 and n is sufficiently large (n ≥ 31 for the lower-uniformity case, n ≥ 55 for the higher-uniformity case). The main theorem splits into two parts: one for r = floor((n-1)/2)-1, handled by a graph-compatibility argument (Theorem 10), and one for r ≥ floor((n-1)/2), handled via the incidence bipartite graph and two structural theorems (Theorems 9 and 8). The proof uses the Kostochka–Luo–McCourt Hamiltonian Berge cycle theorem as a black box, then studies the set of extra edges outside that cycle. Sharpness is witnessed by three explicit constructions whose minimum degrees fall one below the thresholds.

Significance. If the proof is completed, the result is significant: it provides exact Dirac-type thresholds for Berge pancyclicity in the large-uniformity regime, completing the earlier result of Bailey–Li–Luo and giving a hypergraph analogue of Bondy's meta-conjecture. The paper is well structured, and several ingredients are elegant and useful: the shifting-function lemma, the self-shift complementary set structure, and the cycle-compatibility framework. The sharpness constructions are natural and clearly explained. The reliance on an external published theorem is methodologically acceptable, though it means the new result inherits the correctness of that theorem.

major comments (3)

[Section 3, Case 4 (Theorem 9), after Claim 13] The assertion that a k-SSC set contains an interval of d consecutive vertices is not justified and is false in general. For n=24, k=21, d=gcd(24,21)=3, the set A={v0,v2,v4,...,v22} satisfies A∩(k+A)=∅ and |A|=12, hence is k-SSC with 0∈A, but it contains no three consecutive vertices. The subsequent construction of the (k+1)-cycle requires two disjoint intervals of length d inside U'_0, so the proof needs an argument using the specific size-2 case m+p=d+1, not an appeal to the definition of k-SSC.
[Section 3, Case 4, final subcase k=d=n/2] The reduction "Without loss of generality, assume {v0,...,v_{k-1}} = U'_0" is not valid. For k=n/2, a k-SSC set is any transversal of the pairs {i, i+k}; for example with n=8, k=4, {v0,v1,v2,v7} is k-SSC but not a contiguous block. The subsequent edge-counting and Berge-cycle construction rely on contiguity, so this normalization must be proved or the argument must be adapted to arbitrary transversals.
[Section 3, Case 4, paragraph after Claim 12] The step concluding that the index set of U'_0 is k-SSC is missing an argument. The absence of a (k+1)-Berge cycle only rules out k-chords, i.e., pairs at distance k contained in a single extra edge; it does not by itself prevent U'_0 from containing two vertices at distance k in different edges. One must first use |U'_0|=r+1 and |E'_0|≥3 to show that every pair of vertices in U'_0 is contained together in some edge of E'_0, since each edge in E'_0 misses exactly one vertex of U'_0. This is easy to supply but is load-bearing for the SSC conclusion.

minor comments (5)

[Section 2.1] The displayed hamiltonian cycle in the proof of Lemma 11 is written as "C = v0e0v1...vn−1en−1v1"; the final vertex should be v0.
[Section 2, proof of Theorem 5] The sentence "By Theorem 8, if δ(G) ≥ m/3+4, then G is weakly pancyclic with girth 4" should say "weakly bipancyclic" because G is bipartite; the intended conclusion for H is unaffected since only even cycles in G are relevant.
[Section 4, Claims 16 and 17] The notation e_t is reused for an extra edge after having denoted a cycle edge of C; this is confusing and should be renamed, for instance e*.
[Section 1, Construction 1] The sentence "one can check that the minimum degrees of the graphs are one less than those in Theorem 5" should say "hypergraphs" instead of "graphs".
[Section 2, proof of Theorem 5] The inequality r(n/2−15)/n ≥ c_r for n ≥ 55 is asserted tersely; spelling out the two cases r=floor((n−1)/2) and r≥n/2 would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the pancyclicity proof uses an external Hamiltonian-cycle theorem whose assumptions do not include the target.

full rationale

The paper's derivation is not circular. Theorem 5 is proved by first invoking Theorem 3 of Kostochka, Luo, and McCourt to obtain a Hamiltonian Berge cycle under exactly the same degree conditions, and then proving Theorems 9 and 10, which show that a Hamiltonian Berge cycle plus enough extra edges forces Berge cycles of every length from 2 to n. The cited theorem is an external published result whose hypotheses are degree conditions only; it says nothing about pancyclicity, so using it as an input is not equivalent to assuming the conclusion. Although one of the current authors is also an author of [18], this is a case of ordinary reliance on a previously established external theorem, not a self-referential chain: the theorem is parameter-free and its stated assumptions do not include the target result. The complementary small-r pancyclicity theorem from the authors' prior work is cited only to delineate the completed range and is not used in the proof of Theorem 5. No parameter is fitted to the target cycle lengths, and the short-cycle constructions in the proofs of Theorems 9 and 10 are built explicitly from chords in extra edges, shifting functions, and incidence-graph cycles rather than assumed. The sharpness constructions are checked by independent non-Hamiltonicity arguments, not by invoking the main theorem. Thus no claimed prediction or first-principles result reduces, by the paper's own equations or by self-citation, to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

There are no fitted constants or invented entities. Constants such as 5(r-1)+2 and c_r=6 are chosen to make finite inequalities work, not fitted to data. The proof relies on standard external theorems, the main one being the Hamiltonian Berge cycle theorem, plus Hall's theorem and several published pancyclicity results.

assumptions (5)

standard math Theorem 3 of Kostochka, Luo, and McCourt: under the stated minimum degree thresholds, H has a Berge Hamiltonian cycle.
Invoked in Section 2 as the starting cycle C. It is external to this paper and one of its authors overlaps with the present authors.
standard math Brandt's weak pancyclicity theorem: every nonbipartite graph of order n with minimum degree at least (n+2)/3 is weakly pancyclic with girth 3 or 4.
Used in the proof of Theorem 10 to conclude that the auxiliary graph G or G' is pancyclic when it contains a triangle.
standard math Hu-Sun weak bipancyclicity theorem: Hamiltonian bipartite graphs with minimum degree at least n/3+4 are bipancyclic, with a stronger version used as Theorem 8.
Used in Section 2 to translate cycles in the incidence graph into Berge cycles of every length in the hypergraph.
standard math Bondy's pancyclicity theorem: an n-vertex Hamiltonian graph with at least n^2/4 edges is pancyclic or equals K_{n/2,n/2}.
Used in Theorem 10 when the auxiliary graph has minimum degree larger than n/2.
standard math Hall's marriage theorem.
Used in Claims 13 and 15 to find matchings that assign distinct hyperedges to edges of a graph cycle.

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Cite this review

Pith. "Pith review of Pancyclicity in hypergraphs with large uniformity." pith.science (2026). https://pith.science/paper/6BFUK6CO

@misc{pith2026250500130,
  author       = {Pith},
  title        = {Pith review of: Pancyclicity in hypergraphs with large uniformity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6BFUK6CO}},
  note         = {Machine review of arXiv:2505.00130}
}

abstract

A Berge cycle of length $\ell$ in a hypergraph $\mathcal{H}$ is a sequence of alternating vertices and edges $v_0e_0v_1e_1...v_\ell e_\ell v_0$ such that $\{v_i,v_{i+1}\}\subseteq e_i$ for all $i$, with indices taken modulo $\ell$. For $n$ sufficiently large and $r\geq \lfloor\frac{n-1}{2}\rfloor-1$ we prove exact minimum degree conditions for an $n$-vertex, $r$-uniform hypergraph to contain Berge cycles of every length between $2$ and $n$. In conjunction with previous work, this provides sharp Dirac-type conditions for pancyclicity in $r$-uniform hypergraphs for all $3\leq r\leq n$ when $n$ is sufficiently large.

Figures

Figures reproduced from arXiv: 2505.00130 by the authors.

**Figure 1.** An (n − s + 1)-Berge cycle in cases 1 and 2 respectively. 2.2 Self-Shift Complementary Sets When forbidding k-chords for some k ∈ [n − 1], we will often consider sets A ⊂ V (H) where vi ∈ A implies vi+k ∈ A / . If A = {i : vi ∈ A} is the set of indices from A, then A ∩ (k + A) = ∅, where k + A := {k + a (mod n) : a ∈ A}. Moreover, when |A| = n/2, then A = k + A, the complement of k + A in Z/nZ. For A ⊂ Z/nZ and k ∈ … view at source ↗

**Figure 2.** The set Oˆ 0 with n = 20, k = 6. Gray points are in A and white points in A. In the following proposition, we use some elementary group theory. For k ∈ Z/nZ, we denote by hki the subgroup of Z/nZ generated by k, i.e., hki = {ik (mod n) : i ∈ Z}. Then for j ∈ Z/nZ, j + hki is a coset of the subgroup hki in Z/nZ. Proposition 1 (SSC structure). Let k ∈ {1, . . . , n − 1}. Suppose A ⊂ Z/nZ with 0 ∈ A, |A| = n/2 and (k +… view at source ↗

**Figure 3.** Even length Berge cycles. Suppose first that j is even. Let a and b be even integers such that 0 ≤ a < j and 0 ≤ b < n − j. The Berge cycle v0e0vj ...vj−aevn−ben−b+1...v0 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [11]

Edge conditions and cycle st ructure in bipartite graphs

R. C. Entringer and E. Schmeichel. “Edge conditions and cycle st ructure in bipartite graphs”. In: Ars Combinatoria 26 (1988), pp. 229–232

work page 1988

[2] [1]

A condition for a hamiltonian bipartite graph to be bipanc yclic

D. Amar. “A condition for a hamiltonian bipartite graph to be bipanc yclic”. In: Discrete mathematics 102.3 (1992), pp. 221–227

work page 1992

[3] [2]

Bailey, Y

T. Bailey, Y. Li, and R. Luo. Berge Pancyclic hypergraphs. arXiv:2 410.21733

work page

[4] [3]

Weakly pancyclic graphs

B. Bollob` as and A. Thomason. “Weakly pancyclic graphs”. In: Jo urnal of Combinato- rial Theory, Series B 77.1 (1999), pp. 121–137

work page 1999

[5] [4]

Pancyclic graphs I

J.A Bondy. “Pancyclic graphs I”. In: Journal of Combinatorial T heory, Series B 11.1 (1971), pp. 80–84

work page 1971

[6] [5]

Suﬃcient conditions for graphs to contain all subgr aphs of a given type

S. Brandt. “Suﬃcient conditions for graphs to contain all subgr aphs of a given type”. PhD thesis. Freie Universit /dieresis.ts1at Berlin, 1994

work page 1994

[7] [6]

Weakly pancyclic grap hs

S. Brandt, R. Faudree, and W. Goddard. “Weakly pancyclic grap hs”. In: Journal of Graph Theory 27.3 (1998), pp. 141–176. 2W e are unfortunately unable to ﬁnd their paper online. 17

work page 1998

[8] [7]

A rainbow Dirac’s theorem

M. Coulson and G. Perarnau. “A rainbow Dirac’s theorem”. In: SI AM Journal on Discrete Mathematics 34.3 (2020), pp. 1670–1692

work page 2020