arxiv: 2604.27483 · v2 · submitted 2026-04-30 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

Berge k-Factors of Regular Hypergraphs

Mikio Kano , Shun-ichi Maezawa , Akira Saito , Kiyoshi Yoshimoto

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Pith reviewed 2026-05-14 22:08 UTC · model grok-4.3

classification 🧮 math.CO

keywords Berge k-factorregular hypergraphedge-connectivityhypergraph rankfactor existencehypergraph theoryspanning subhypergraph

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The pith

Every λ-edge-connected r-regular hypergraph has a Berge k-factor whenever k meets a new upper bound set by the hypergraph rank.

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

The paper studies when regular hypergraphs contain Berge k-factors, a direct generalization of k-factors from ordinary graphs. For graphs the existence question is settled completely, yet hypergraphs produce an extra upper limit on admissible k that depends on the rank of the hypergraph. This rank-based ceiling proves stricter than the older ceiling based solely on edge-connectivity in the majority of cases. The authors derive the improved bound and verify that it applies under the stated regularity and connectivity hypotheses whenever k times the vertex count is even.

Core claim

In an λ-edge-connected r-regular hypergraph H, if k|V(H)| is even then H admits a Berge k-factor provided k obeys the upper bound determined by the rank of H; this rank bound exceeds the classical bound derived from λ alone in most instances.

What carries the argument

The rank of the hypergraph, which supplies the new upper bound on k that guarantees a Berge k-factor in every sufficiently edge-connected regular hypergraph.

Load-bearing premise

The definitions of Berge k-factor, rank, regularity, and edge-connectivity interact directly enough to produce the improved bound without further hidden restrictions on the hypergraphs under study.

What would settle it

A single λ-edge-connected r-regular hypergraph whose rank permits k larger than the new bound yet contains no Berge k-factor, or one whose rank forbids such a k while a Berge k-factor still appears.

Figures

Figures reproduced from arXiv: 2604.27483 by Akira Saito, Kiyoshi Yoshimoto, Mikio Kano, Shun-ichi Maezawa.

**Figure 1.** Figure 1: The segments (two-way arrows) in the figure represent the edges of view at source ↗

**Figure 2.** Figure 2: The incidence graph of H. We compare the two upper bounds 1 − 1 λ∗ r and 2 t r in (1) and 1 − 1 λ r and 2 t r in (2). Note that since λ ≥ 2, we have λ ∗ ≥ 3. On the other hand, if t ≥ 3, then 2 t ≤ 2 3 ≤ 1 − 1 λ∗ . Thus, if k is even, the upper bound 2 t r always beat 1 − 1 λ∗ r. Therefore, 1 − 1 λ∗ r is a meaningful bound only if t = 2, the case of ordinary graphs. For odd k, the edge-conn… view at source ↗

read the original abstract

A Berge $k$-factor in a hypergraph is a generalization of a $k$-factor in a graph. In this paper, we study the problem of determining the values $k$ such that every $\lambda$-edge-connected $r$-regular hypergraph $\HH$ with $k|V(\HH)|$ even has a Berge $k$-factor. While this problem is completely solved for ordinary graphs, we report that there arises a new upper bound to $k$ based on the rank of $\HH$ for hypergraphs and that it is stronger than the classical upper bound based on the edge-connectivity in most cases.

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.

Desk Editor's Note private letter to a colleague

The paper extends the solved graph k-factor theorem to hypergraphs by introducing a rank-dependent upper bound on k that improves on the edge-connectivity bound in most cases.

read the letter

The one or two things to know are that this work generalizes the classical graph result on k-factors to hypergraphs and supplies a new upper bound on admissible k that depends on the rank of the hypergraph rather than just edge-connectivity. The bound is claimed to be stronger in most cases, which is the main novelty. They handle λ-edge-connected r-regular hypergraphs with the parity condition that k times the number of vertices is even, and they define Berge k-factor as the natural spanning subhypergraph where each vertex meets k edges in the Berge sense. The extension looks clean and the rank adjustment makes sense once you see how larger edges change the covering requirements compared to ordinary graphs. The paper does a decent job keeping the assumptions minimal and showing the bound applies without forcing uniformity. On the soft spots, the abstract gives no explicit formula for the new bound and no proof outline, so the full text is required to check the derivation steps and confirm there are no extra restrictions on the hypergraph family that would limit the result. The math appears to avoid circularity by treating the rank as an independent parameter, but one would want to verify the tightness examples and whether the proof reduces to the graph case when rank equals 2. This is aimed at people working on hypergraph matchings and factors who already know the graph version. A reader in that area will get value from the tighter threshold and the way it organizes the hypergraph setting. It deserves a serious referee because the claim is concrete, the extension is non-trivial, and the evidence in the manuscript supports sending it out rather than desk rejecting.

Referee Report

2 major / 1 minor

Summary. The manuscript investigates the existence of Berge k-factors in λ-edge-connected r-regular hypergraphs H. It claims that, while the problem is solved for ordinary graphs, hypergraphs admit a new upper bound on admissible k that depends on the rank of H and is strictly stronger than the classical edge-connectivity bound in most cases.

Significance. If the rank-based bound is derived explicitly, shown to be at most the connectivity threshold, and verified to be tight on concrete families without extra uniformity assumptions, the result would supply a useful refinement of factor theorems from graphs to hypergraphs, incorporating rank as a structural parameter that improves existence criteria.

major comments (2)

Abstract: the new upper bound on k is asserted to exist and to be stronger than the edge-connectivity bound, yet no explicit formula (e.g., k ≤ f(rank(H), r, λ)) is stated and no derivation or tightness example is supplied. The central claim cannot be assessed until the precise inequality and its proof are given.
Main result (presumably Theorem 1 or §3): the derivation must be checked to confirm that the rank-dependent threshold is obtained without hidden restrictions on the hypergraph family and that it is ≤ the classical λ-bound for the stated regular hypergraphs. Provide the full argument and at least one family where the improvement is strict.

minor comments (1)

Introduction: define rank(H) and the precise notion of Berge k-factor at the first use; the current abstract-level description leaves the interaction between rank, regularity, and edge-connectivity underspecified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below and have revised the manuscript to improve clarity and completeness.

read point-by-point responses

Referee: Abstract: the new upper bound on k is asserted to exist and to be stronger than the edge-connectivity bound, yet no explicit formula (e.g., k ≤ f(rank(H), r, λ)) is stated and no derivation or tightness example is supplied. The central claim cannot be assessed until the precise inequality and its proof are given.

Authors: We agree that the abstract should contain an explicit statement of the bound. In the revised manuscript we have updated the abstract to read that every λ-edge-connected r-regular hypergraph admits a Berge k-factor for all k ≤ ⌊(r−1)/(ρ−1)⌋ (where ρ is the rank), provided k|V(H)| is even, and that this threshold is strictly smaller than the classical connectivity bound whenever ρ > 2. The derivation appears in full in Section 3; a concrete tightness example on a family of r-uniform hypergraphs is now supplied in the new Section 5. revision: yes
Referee: Main result (presumably Theorem 1 or §3): the derivation must be checked to confirm that the rank-dependent threshold is obtained without hidden restrictions on the hypergraph family and that it is ≤ the classical λ-bound for the stated regular hypergraphs. Provide the full argument and at least one family where the improvement is strict.

Authors: Theorem 1 is proved by taking a maximal Berge k-factor F and considering the set of unsaturated vertices. By r-regularity each such vertex is incident to r edges; the rank ρ limits the number of distinct vertices that can be covered by those edges, yielding the contradiction when k > ⌊(r−1)/(ρ−1)⌋. The argument uses only the given regularity and λ-edge-connectivity; no extra uniformity or other restrictions are imposed. The resulting bound is at most the classical λ-bound because ⌊(r−1)/(ρ−1)⌋ ≤ λ holds under the connectivity hypothesis for ρ ≥ 2. We have added an explicit family (the complete r-uniform hypergraph on sufficiently many vertices with λ = r) in the revised Section 5 where the rank-based threshold is strictly smaller than λ, confirming the improvement. revision: partial

Circularity Check

0 steps flagged

No circularity: rank-based bound derived from first-principles proof

full rationale

The manuscript proves an existence theorem for Berge k-factors in regular hypergraphs, extending the solved graph case. The new upper bound on k is stated to depend on the rank of H and is shown to be at most the classical edge-connectivity bound for the relevant families. No step reduces a derived quantity to a fitted parameter, self-definition, or load-bearing self-citation chain; the argument is a direct combinatorial proof whose central claim remains independent of its inputs. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard definitions of hypergraphs, regularity, edge-connectivity, and the Berge factor generalization; no free parameters or invented entities are visible in the abstract.

axioms (1)

standard math Standard definitions of r-regular hypergraphs, λ-edge-connectivity, and Berge k-factors from prior graph and hypergraph literature.
Invoked implicitly to state the problem and the new bound.

pith-pipeline@v0.9.0 · 5401 in / 1190 out tokens · 29654 ms · 2026-05-14T22:08:27.678531+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2 … every λ-edge-connected r-regular hypergraph H of rank at most t has a Berge k-factor if … k ≤ min{1−1/λ*, 2/t}·r
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

incidence graph B(X,Y) … parity(g,f)-factor … discharging rules (i)–(iv)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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