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arxiv: 2604.14978 · v1 · submitted 2026-04-16 · 🧮 math.CO

Counting tight Hamilton cycles in Dirac hypergraphs

Felix Joos , Xinyue Xie This is my paper

Pith reviewed 2026-05-10 10:53 UTC · model grok-4.3

classification 🧮 math.CO
keywords tight Hamilton cyclesk-uniform hypergraphsminimum codegreehypergraph entropyfractional matchingscounting Hamilton cyclesDirac hypergraphs
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The pith

Hypergraph entropy supplies a tight lower bound on the number of tight Hamilton cycles in k-uniform hypergraphs with codegree condition δ > 1/2.

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

The paper establishes a lower bound on the logarithm of the number of tight Hamilton cycles in any k-uniform hypergraph where every (k-1)-subset lies in at least δn edges for δ greater than one half. The bound is stated in terms of the hypergraph entropy h(G), a quantity computed from the sizes of perfect fractional matchings. If the bound holds, it matches the expected count in nearly regular hypergraphs of the same density and immediately yields that the total number of cycles is at least (δ minus a vanishing term) to the power n times n factorial. A reader would care because the result gives an explicit, entropy-based counting tool for a canonical extremal problem in hypergraph combinatorics.

Core claim

For a k-uniform hypergraph G on n vertices with minimum codegree at least δn where δ exceeds 1/2, log Ψ(G) is at least k h(G) minus n log of the binomial coefficient n choose k-1, plus n log n, minus n log e, up to an o(n) error term. Here Ψ(G) counts the tight Hamilton cycles and h(G) is the hypergraph entropy defined from perfect fractional matchings. The inequality is asymptotically tight for all nearly regular hypergraphs and, in particular, for the binomial random hypergraph; it also implies the lower bound Ψ(G) at least (δ minus o(1)) to the power n times n factorial.

What carries the argument

Hypergraph entropy h(G), defined via the maximum size of a perfect fractional matching and used to quantify the effective density available for cycle counting.

If this is right

  • The stated lower bound on log Ψ(G) is asymptotically tight for every nearly regular hypergraph.
  • The same precision holds for the binomial random hypergraph with edge probability above the threshold.
  • The number of tight Hamilton cycles satisfies Ψ(G) at least (δ minus o(1)) to the power n times n factorial.

Where Pith is reading between the lines

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

  • The entropy technique may adapt to count loose Hamilton cycles or other spanning structures once suitable fractional matchings are identified.
  • If weaker degree conditions still permit perfect fractional matchings, the same lower bound could apply beyond the Dirac regime.
  • The approach supplies a template for asymptotic enumeration of other subgraphs whose existence is controlled by fractional matching numbers.

Load-bearing premise

The minimum codegree condition δ greater than 1/2 is required to guarantee the existence of perfect fractional matchings that define the entropy and support the counting argument.

What would settle it

A concrete k-uniform hypergraph on sufficiently large n with every (k-1)-subset in more than n/2 edges but possessing strictly fewer than (δ minus 0.01) to the power n times n factorial tight Hamilton cycles would disprove the bound.

read the original abstract

Suppose $G$ is a $k$-uniform hypergraph on $n$ vertices such that every $(k-1)$-subset $S$ of $V(G)$ belongs to at least $\delta n$ edges, where $\delta> 1/2$. Let $\Psi(G)$ denote the number of tight Hamilton cycles in $G$, that is, cyclic orderings of $V(G)$ in which every $k$ consecutive vertices form an edge. We prove that $\log\Psi(G)\ge kh(G)-n\log{n\choose k-1}+n\log n-n\log e-o(n)$, where $h(G)$ is the hypergraph entropy of $G$, defined via perfect fractional matchings. This bound is tight, for example, for all (nearly) regular hypergraphs, in particular for the binomial random hypergraph. It also implies a conjecture by Ferber, Hardiman and Mond, stating that $\Psi(G)\ge (\delta-o(1))^n n!$.

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 paper proves that for a k-uniform hypergraph G on n vertices with minimum codegree at least δn (δ > 1/2), the number Ψ(G) of tight Hamilton cycles satisfies log Ψ(G) ≥ k h(G) − n log binom(n, k−1) + n log n − n log e − o(n), where h(G) is the hypergraph entropy defined via perfect fractional matchings. The bound is asserted to be tight for nearly regular hypergraphs (including the random case) and implies the Ferber-Hardiman-Mond conjecture that Ψ(G) ≥ (δ − o(1))^n n!.

Significance. If the derivation holds, the result supplies a precise entropy-based lower bound on tight Hamilton cycles under a Dirac-type codegree condition, resolving a stated conjecture and providing a template for counting problems in hypergraphs via fractional matching polytopes. The tightness claim for regular instances and the o(n) error control would be notable strengths in extremal combinatorics.

minor comments (3)
  1. The abstract and introduction should explicitly state the range of k for which the result holds (e.g., fixed k or k = o(n)); the entropy definition via fractional matchings appears only in the abstract and would benefit from a self-contained paragraph in §2.
  2. The o(n) error term is stated without an explicit dependence on k or δ; adding a remark on how the concentration arguments in the counting step produce this term would improve readability.
  3. The tightness statement for the binomial random hypergraph is asserted but not accompanied by a short calculation or reference to the entropy value in that model; a one-line verification would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of its significance in providing an entropy-based lower bound on tight Hamilton cycles under the Dirac codegree condition and its resolution of the Ferber-Hardiman-Mond conjecture. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines the hypergraph entropy h(G) externally via perfect fractional matchings (a standard concept from the fractional matching polytope), then derives the lower bound on log Ψ(G) from this definition using the codegree condition δ > 1/2 to guarantee the matchings exist and standard concentration arguments for the counting step. The bound is shown tight for nearly regular cases including the random hypergraph, but this is a verification of tightness rather than a reduction of the result to a fitted parameter or self-referential input. No step reduces by the paper's own equations to a self-definition, fitted prediction, or load-bearing self-citation chain; the derivation remains self-contained against external matching theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition of hypergraph entropy via fractional matchings and standard asymptotic combinatorics; no new entities or fitted constants are introduced.

axioms (2)
  • domain assumption Existence of perfect fractional matchings in k-uniform hypergraphs satisfying the Dirac minimum codegree condition δ > 1/2.
    Invoked to define h(G) and to obtain the entropy term in the bound.
  • standard math Standard properties of logarithms, binomial coefficients, and o(n) error terms in counting arguments.
    Used throughout the derivation of the logarithmic inequality.

pith-pipeline@v0.9.0 · 5464 in / 1393 out tokens · 39864 ms · 2026-05-10T10:53:01.390625+00:00 · methodology

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Reference graph

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