Hamiltonian Cycles in Simplicial and Supersolvable Hyperplane Arrangements

Jan Stricker; Jasmin Walizadeh; Tobias Schnieders; Veronika K\"orber

arxiv: 2508.14538 · v3 · submitted 2025-08-20 · 🧮 math.CO

Hamiltonian Cycles in Simplicial and Supersolvable Hyperplane Arrangements

Veronika K\"orber , Tobias Schnieders , Jan Stricker , Jasmin Walizadeh This is my paper

Pith reviewed 2026-05-18 22:37 UTC · model grok-4.3

classification 🧮 math.CO

keywords Hamiltonian cycleshyperplane arrangementssupersolvable arrangementstope graphsoriented matroidssimplicial arrangementsreflection arrangements

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The pith

All supersolvable hyperplane arrangements and supersolvable oriented matroids have Hamiltonian cycles in their tope graphs.

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

The paper establishes that supersolvable hyperplane arrangements and supersolvable oriented matroids possess Hamiltonian cycles in their tope graphs. The proof is constructive and relies on the inductive definition of supersolvability to build the cycle step by step from smaller subarrangements. This result builds on earlier findings for reflection arrangements and includes a computational verification for all three-dimensional simplicial arrangements in a known catalogue. A reader would care because such cycles correspond to Gray-code-like orderings of the regions, which can be useful in enumeration, optimization, and understanding the combinatorial structure of these geometric objects. If the claim holds, it provides a systematic way to traverse all regions of these arrangements without repetition.

Core claim

We prove that all supersolvable hyperplane arrangements have Hamiltonian cycles in their tope graphs by a constructive method based on their inductive structure. The same holds for supersolvable oriented matroids. Additionally, we confirm Hamiltonicity for all 3-dimensional simplicial arrangements from the Grünbaum-Cuntz catalogue and extend previous results to show that all restrictions of finite reflection arrangements, including Weyl groupoids and crystallographic arrangements, admit such cycles.

What carries the argument

The inductive structure of supersolvable arrangements, which permits a recursive construction of the Hamiltonian cycle by extending cycles from smaller arrangements when adding a new hyperplane.

If this is right

Every supersolvable oriented matroid admits a Hamiltonian cycle in its tope graph.
All restrictions of finite reflection arrangements, including Weyl groupoids and crystallographic arrangements, have Hamiltonian cycles.
The construction yields an explicit algorithm for producing the cycle in any supersolvable case.
All 3-dimensional simplicial arrangements in the Grünbaum-Cuntz catalogue have Hamiltonian cycles.

Where Pith is reading between the lines

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

The inductive method may extend to other classes of arrangements or matroids defined by recursive deletion or restriction operations.
Such cycles could yield efficient traversal orders for region enumeration in geometric combinatorics beyond the supersolvable setting.
The result suggests testing Hamiltonicity in tope graphs of arrangements that share partial inductive properties with supersolvable ones.

Load-bearing premise

The inductive structure of supersolvable arrangements can be used to build the cycle recursively without getting stuck when extending from smaller to larger arrangements.

What would settle it

A specific supersolvable hyperplane arrangement or supersolvable oriented matroid whose tope graph contains no Hamiltonian cycle, or a case where the recursive extension step fails to produce a valid cycle.

read the original abstract

Motivated by the Gray code interpretation of Hamiltonian cycles in Cayley graphs, we investigate the existence of Hamiltonian cycles in tope graphs of hyperplane arrangements, with a focus on simplicial, reflection, and supersolvable arrangements. We confirm Hamiltonicity for all 3-dimensional simplicial arrangements listed in the Gr\"unbaum--Cuntz catalogue. Extending earlier results by Conway, Sloane, and Wilks, we prove that all restrictions of finite reflection arrangements, including all Weyl groupoids and crystallographic arrangements, admit Hamiltonian cycles. Finally, we further establish that all supersolvable hyperplane arrangements and supersolvable oriented matroids have Hamiltonian cycles, offering a constructive proof based on their inductive structure.

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

Constructive inductive proof that supersolvable arrangements and matroids have Hamiltonian tope-graph cycles, plus a catalogue check for 3D simplicial cases.

read the letter

The main thing here is a constructive proof that every supersolvable hyperplane arrangement and supersolvable oriented matroid has a Hamiltonian cycle in its tope graph, built recursively from the inductive decomposition. They also verify the property for all 3-dimensional simplicial arrangements in the Grünbaum-Cuntz list and extend the earlier Conway-Sloane-Wilks results to all restrictions of reflection arrangements, including Weyl groupoids and crystallographic ones. The argument uses the supersolvable structure to extend a cycle on a restriction by inserting new topes along the added hyperplane while keeping adjacency, with a linear extension order that prevents isolated topes. Base cases are handled by direct enumeration. This is a clear step forward from the reflection case and gives an explicit method rather than a non-constructive existence argument. The 3D simplicial part is solid but limited to a finite check, so it confirms the pattern without proving a broader theorem. The induction looks to cover the necessary cases without obvious missing steps or extra assumptions, though the full write-up would need a close read for any edge cases in the insertion process. The paper stays within standard definitions and builds directly on cited prior results with no circularity. Readers working in combinatorial geometry, hyperplane arrangements, or oriented matroids will get the most out of the constructive technique and the explicit examples. It has enough new, grounded content to deserve a serious referee who can verify the inductive details and the catalogue enumeration.

Referee Report

0 major / 2 minor

Summary. The manuscript proves the existence of Hamiltonian cycles in tope graphs of hyperplane arrangements, verifying the property for all 3-dimensional simplicial arrangements in the Grünbaum-Cuntz catalogue, extending earlier results of Conway, Sloane, and Wilks to prove it for all restrictions of finite reflection arrangements (including Weyl groupoids and crystallographic arrangements), and establishing it for supersolvable hyperplane arrangements and supersolvable oriented matroids via an explicit constructive inductive proof based on their decomposition into smaller arrangements.

Significance. If the results hold, this constitutes a solid contribution to the combinatorics of arrangements and oriented matroids by confirming Hamiltonicity for several natural classes and supplying a constructive method. The explicit inductive construction, with base cases verified by direct enumeration and the inductive step extending a cycle on a restriction by inserting new topes along the added hyperplane while preserving adjacency, is a clear strength; the argument that the supersolvable decomposition guarantees a linear extension order without isolated topes is presented without apparent internal inconsistency.

minor comments (2)

[Abstract] Abstract: the motivation linking Hamiltonian cycles to Gray codes in Cayley graphs is stated, but a single sentence indicating how the supersolvable inductive construction yields an explicit Gray-code-like ordering would strengthen the opening.
[Final section] Final section: the claim that the linear extension order never leaves an isolated tope is load-bearing for the inductive step; a one-sentence reminder of the precise supersolvability property (e.g., the existence of a modular element) used to guarantee adjacency would improve readability without lengthening the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the constructive inductive proof as a strength, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; explicit inductive construction is self-contained

full rationale

The paper supplies a direct constructive proof of Hamiltonicity for supersolvable hyperplane arrangements and oriented matroids via induction on their standard inductive decomposition. Base cases are checked by explicit enumeration of small arrangements, and the inductive step inserts new topes along the added hyperplane while preserving adjacency using the linear extension order guaranteed by supersolvability. This relies only on the definition of supersolvability and the tope graph, plus external prior results on reflection arrangements by Conway-Sloane-Wilks; no equation or claim reduces to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivation therefore stands independently of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard axiomatic definition of hyperplane arrangements and the inductive characterization of supersolvability; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption Supersolvable arrangements admit an inductive decomposition into smaller arrangements along a modular element.
Invoked to obtain the recursive construction of the Hamiltonian cycle.

pith-pipeline@v0.9.0 · 5649 in / 1113 out tokens · 31501 ms · 2026-05-18T22:37:04.994731+00:00 · methodology

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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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

We prove the statement by induction on n = rk(A). For n = 2, all hyperplane arrangements trivially have a Hamiltonian cycle. ... We can traverse all fibers by P+(B1), P−(B2), P+(B3), …, P−(B2k) … because A0 has an even number of regions.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat induction echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

An arrangement A of rank n ≥ 3 is supersolvable if and only if it can be written as a disjoint union of arrangements A = A0 ⊎ A1 … where A0 is a supersolvable arrangement of rank n−1

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

On gamma-vectors and Chow polynomials of restrictions of reflection arrangements
math.CO 2025-11 unverdicted novelty 6.0

All restrictions of reflection arrangements are gamma-positive; type B Chow polynomials admit an explicit combinatorial formula, and intermediate type D restrictions interpolate arithmetically between B and D invariants.