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arxiv: 2606.12230 · v2 · pith:R4MEPR3Tnew · submitted 2026-06-10 · 🧮 math.CO

One extra edge forces Berge pancyclicity

Henry Shin This is my paper

Pith reviewed 2026-06-27 09:04 UTC · model grok-4.3

classification 🧮 math.CO
keywords Berge cycleshypergraph pancyclicityHamiltonian Berge cycler-uniform hypergraphsedge-reassignmentsum-free stabilityadditive covering
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The pith

Any Hamiltonian Berge cycle plus one extra edge yields Berge cycles of every length from 2 to n in large r-uniform hypergraphs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for all sufficiently large n, with r equal to floor of (n-1)/2, the edges belonging to any Hamiltonian Berge cycle in a simple n-vertex r-uniform hypergraph, when joined by any single additional edge, produce Berge cycles of every length between 2 and n inclusive. This directly answers a question posed by Bailey, Hollars, Li and Luo about the minimal extra structure needed to guarantee full length coverage. A reader would care because it shows that hypergraph cycle systems become pancyclic with almost no additional edges beyond a Hamiltonian one, mirroring but extending classical graph pancyclicity results. The argument splits into odd-order and even-order cases, each using tailored combinatorial arguments to cover all lengths.

Core claim

We prove that the edges of any Hamiltonian Berge cycle in a simple n-vertex r-uniform hypergraph, together with any one additional edge, contain Berge cycles of every length from 2 to n, for all sufficiently large n with r = floor((n-1)/2). In odd order we prove a stronger prescribed-unused-edge theorem using rigidity of large subsets of odd cyclic groups and an alternating matching exchange. In even order we introduce a two-gap edge-reassignment method. Split locks cover all lengths outside a seven-term middle band. The absence of the central length forces an exact reflected translation-wave structure, which is eliminated by an additive covering theorem derived from sum-free stability. The

What carries the argument

the edges of a Hamiltonian Berge cycle together with one additional edge in an r-uniform hypergraph

If this is right

  • Berge cycles of every length from 2 to n exist within the given edges.
  • In the odd-order case the result continues to hold even when one specific edge is required to remain unused.
  • Lengths outside a central band of seven consecutive values are covered by split-lock constructions.
  • The central length, if missing, produces a reflected translation-wave structure that is ruled out by the additive covering theorem.
  • Near-central lengths are obtained from a two-defect recurrence combined with bounded-run forcing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same one-extra-edge principle might hold for smaller values of n that can be checked by computer search.
  • The stability methods drawn from sum-free sets could be reused for other minimal-edge problems in hypergraph cycle theory.
  • Analogous statements may apply to non-uniform hypergraphs or to directed versions of Berge cycles.
  • The result indicates that Berge pancyclicity is a robust property once a single extra edge is present, potentially informing constructions in design theory.

Load-bearing premise

n must be large enough that the rigidity of large subsets of odd cyclic groups and the matching exchange methods hold without exception.

What would settle it

An r-uniform hypergraph on some sufficiently large n, containing a Hamiltonian Berge cycle and one extra edge, that misses a Berge cycle of some length k with 2 less than or equal to k less than or equal to n would serve as a counterexample.

read the original abstract

We resolve a question of Bailey, Hollars, Li and Luo. For all sufficiently large $n$, let $r=\lfloor(n-1)/2\rfloor$. We prove that the edges of any Hamiltonian Berge cycle in a simple $n$-vertex $r$-uniform hypergraph, together with any one additional edge, contain Berge cycles of every length from $2$ to $n$. In odd order we prove a stronger prescribed-unused-edge theorem using rigidity of large subsets of odd cyclic groups and an alternating matching exchange. In even order we introduce a two-gap edge-reassignment method. Split locks cover all lengths outside a seven-term middle band. The absence of the central length forces an exact reflected translation-wave structure, which is eliminated by an additive covering theorem derived from sum-free stability. The remaining near-central lengths follow from a two-defect recurrence and bounded-run forcing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that for all sufficiently large n, with r = floor((n-1)/2), the edges of any Hamiltonian Berge cycle in a simple n-vertex r-uniform hypergraph, together with any one additional edge, contain Berge cycles of every length from 2 to n. The argument splits into odd-order and even-order cases. In odd order a stronger prescribed-unused-edge theorem is proved via rigidity of large subsets of odd cyclic groups and alternating matching exchange. In even order a two-gap edge-reassignment method is introduced. Split locks handle lengths outside a seven-term middle band; the absence of the central length forces a reflected translation-wave structure eliminated by an additive covering theorem derived from sum-free stability; near-central lengths follow from a two-defect recurrence and bounded-run forcing.

Significance. The result resolves a question posed by Bailey, Hollars, Li and Luo. It supplies a self-contained combinatorial proof that combines tools from additive combinatorics (sum-free stability) with direct hypergraph arguments (rigidity, matching exchange, gap reassignment). The stronger odd-order statement with a prescribed unused edge is a notable strengthening. The architecture is conditioned on sufficiently large n, which is the standard setting for the invoked stability and rigidity lemmas.

minor comments (3)
  1. The abstract states that the additive covering theorem is 'derived from sum-free stability' but does not indicate whether the derivation is self-contained or cites an external result; a brief sentence clarifying this would aid readers.
  2. The seven-term middle band is mentioned without an explicit numerical range; stating the precise interval (e.g., lengths k with |k - n/2| ≤ 3) would make the split-lock argument easier to follow.
  3. The manuscript uses the phrase 'for all sufficiently large n' repeatedly; an explicit lower bound on n (even if large) or a remark on how the constants arise from the stability lemmas would strengthen the statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. No major comments appear in the report, so we have no specific points requiring response or revision.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained combinatorial case analysis

full rationale

The paper presents a direct proof via case splits (split locks outside a central band, reflected translation-wave elimination via an additive covering theorem, two-defect recurrence for near-central lengths), with separate handling for odd and even order using rigidity of odd cyclic groups, alternating matching exchange, and two-gap reassignment. These steps are conditioned on sufficiently large n for external stability and rigidity lemmas to apply, but no load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain. The additive covering theorem is derived from sum-free stability (standard external input), and the central claim does not rename or presuppose its own output. No equations or definitions are shown to be equivalent by fiat. This is the normal honest outcome for a self-contained combinatorial argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard mathematical background in group theory and hypergraph combinatorics; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of cyclic groups, matchings, and sum-free sets in additive combinatorics
    Invoked for rigidity of odd cyclic groups, alternating matching exchange, and additive covering theorem.

pith-pipeline@v0.9.1-grok · 5663 in / 1147 out tokens · 22388 ms · 2026-06-27T09:04:18.903266+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 15 canonical work pages

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