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arxiv: 2306.12713 · v2 · submitted 2023-06-22 · 🧮 math.CO

A constructive solution to the Oberwolfach Problem with a large cycle

Tommaso Traetta This is my paper

Pith reviewed 2026-05-24 08:49 UTC · model grok-4.3

classification 🧮 math.CO
keywords Oberwolfach problem2-factorizationamalgamation-detachmentcycleautomorphism groupcomplete graph2-regular graph
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0 comments X

The pith

The Oberwolfach problem has solutions for every 2-regular graph F that contains a cycle longer than an explicit lower bound.

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

The paper constructs solutions to the Oberwolfach problem OP(F) for 2-regular graphs F on v vertices that contain a cycle of length exceeding a specific bound. This is achieved by merging the amalgamation-detachment method with techniques for 2-factorizations whose automorphism groups act nearly regularly on the vertex set. A sympathetic reader cares because this addresses a long-standing open problem from 1967 by providing explicit constructions for many previously unresolved cases, applying to both odd and even v.

Core claim

For every 2-regular graph F of order v, the Oberwolfach problem OP(F) asks whether there is a 2-factorization of K_v (v odd) or K_v minus a 1-factor (v even) into copies of F. We construct solutions to OP(F) whenever F contains a cycle of length greater than an explicit lower bound. Our constructions combine the amalgamation-detachment technique with methods aimed at building 2-factorizations with an automorphism group having a nearly-regular action on the vertex-set.

What carries the argument

Amalgamation-detachment technique combined with 2-factorizations having automorphism groups with nearly-regular action on the vertex set.

If this is right

  • OP(F) is solved constructively for all 2-regular F with a cycle longer than the bound.
  • The method works for both complete graphs K_v and K_v minus a 1-factor depending on the parity of v.
  • The constructions rely on specific automorphism properties to enable the detachment step.
  • This resolves the problem for graphs containing large cycles explicitly.

Where Pith is reading between the lines

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

  • The bound on cycle length might be improvable with refinements to the automorphism constructions.
  • Similar combination of techniques could address other open factorization problems in graph theory.
  • Graphs with multiple long cycles or specific structures might admit even simpler decompositions.

Load-bearing premise

The amalgamation-detachment technique can be combined with the construction of 2-factorizations whose automorphism group has a nearly-regular action on the vertex set.

What would settle it

A concrete 2-regular graph F with a cycle of length above the paper's explicit bound for which the described construction fails to produce a 2-factorization.

read the original abstract

For every $2$-regular graph $F$ of order $v$, the Oberwolfach problem $OP(F)$ asks whether there is a $2$-factorization of $K_v$ ($v$ odd) or $K_v$ minus a $1$-factor ($v$ even) into copies of $F$. Posed by Ringel in 1967 and extensively studied ever since, this problem is still open. In this paper we construct solutions to $OP(F)$ whenever $F$ contains a cycle of length greater than an explicit lower bound. Our constructions combine the amalgamation-detachment technique with methods aimed at building $2$-factorizations with an automorphism group having a nearly-regular action on the vertex-set.

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 / 2 minor

Summary. The manuscript claims to provide explicit constructions solving the Oberwolfach problem OP(F) for every 2-regular graph F of order v that contains at least one cycle longer than an explicit lower bound. The approach combines the amalgamation-detachment technique with constructions of 2-factorizations whose automorphism groups act nearly regularly on the vertex set.

Significance. If the constructions hold, the result would resolve a large and previously open class of instances of a problem posed in 1967, supplying constructive rather than existential solutions together with an explicit cycle-length threshold. The combination of amalgamation-detachment with controlled automorphism actions is a technically substantive contribution that could serve as a template for further cases.

minor comments (2)
  1. The abstract refers to an 'explicit lower bound' on cycle length; the introduction or main theorem statement should state the precise numerical bound (e.g., v/2 or similar) so that the scope of the result is immediately verifiable.
  2. The final sentence of the abstract mentions the 'nearly-regular action'; a short paragraph early in the paper should recall the precise definition of 'nearly regular' used here, including the permitted deviation from regularity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained and constructive

full rationale

The paper claims explicit constructions for solutions to OP(F) when F contains a cycle exceeding an explicit length bound, obtained by combining the amalgamation-detachment technique with 2-factorizations admitting nearly-regular automorphism actions. No load-bearing step reduces a claimed prediction or existence result to a fitted parameter, self-definition, or self-citation chain; the abstract and described method present an independent constructive argument whose validity rests on the explicit techniques rather than on renaming or circular reuse of its own outputs. This is the normal non-circular outcome for a paper whose central claim is a direct construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no free parameters, invented entities, or non-standard axioms are visible. The work rests on standard assumptions of graph theory and factorization existence.

axioms (1)
  • standard math Standard definitions and existence conditions for 2-factorizations of complete graphs (or complete graphs minus a 1-factor).
    Invoked throughout the abstract as the setting for OP(F).

pith-pipeline@v0.9.0 · 5641 in / 1176 out tokens · 24648 ms · 2026-05-24T08:49:17.190873+00:00 · methodology

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Reference graph

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