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arxiv: 2607.01130 · v1 · pith:MFZPMNCXnew · submitted 2026-07-01 · 🧮 math.CO

A complete solution to the generalized honeymoon Oberwolfach problem with one round table

Masoomeh Akbari This is my paper

Pith reviewed 2026-07-02 10:07 UTC · model grok-4.3

classification 🧮 math.CO
keywords honeymoon Oberwolfach problemOberwolfach problemseating arrangementsgraph decompositionscombinatorial designsround tables
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The pith

The obvious necessary conditions for the generalized honeymoon Oberwolfach problem with one round table are also sufficient.

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

The paper provides a complete solution to the generalized honeymoon Oberwolfach problem when there is one round table. It proves that for HOP(2^{}, 2m), where 2n participants from n couples are seated at s tables of size 2 and one table of size 2m, the standard divisibility conditions on the parameters are sufficient to guarantee a schedule where each sits next to their spouse every night and next to every other participant exactly once over the nights. This matters because it resolves the existence question for this specific case of the problem in combinatorial scheduling.

Core claim

The obvious necessary conditions for HOP(2^{<s>}, 2m) to have a solution are also sufficient, giving a complete solution to the generalized HOP with one round table.

What carries the argument

The demonstration that the divisibility requirements on n, s, and m suffice for the existence of the required decompositions or seating schedules.

If this is right

  • If the conditions hold then a complete seating schedule exists for any such s and m.
  • The problem is settled for the case of exactly one round table.
  • Similar sufficiency may hold for variants with additional round tables.

Where Pith is reading between the lines

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

  • This result may extend to cases with multiple round tables if similar techniques apply.
  • It connects the honeymoon variant to the standard Oberwolfach problem solutions.
  • Further work could check if the conditions remain sufficient when table sizes vary more freely.

Load-bearing premise

The divisibility conditions previously identified as necessary are in fact the only barriers to existence.

What would settle it

Finding values of s and m where the divisibility conditions hold but no valid seating schedule exists would disprove the sufficiency claim.

Figures

Figures reproduced from arXiv: 2607.01130 by Masoomeh Akbari.

Figure 1
Figure 1. Figure 1: Replacing the edge of difference d ∗ with two edges of differences di−1 and 1. Lemma 6.4 serves as a reduction step, showing that to find an HOP (Cm)-decomposition of 4K• n for even m and all even n ≥ m, it suffices to find such a decomposition for even n in the interval m ≤ n < 2m. The reduction step for the case of odd n is stated in Lemma 6.6. These lemmas extend the approach from [3], originally develo… view at source ↗
Figure 2
Figure 2. Figure 2: Starter (Cm)-subgraphs for an HOP decomposition of 4K• m+1 with m ≡ 2 (mod 4). is a (Cm)-decomposition for 4K• m+1. Notice that F1, F2, F3, F4 jointly contain exactly one edge from each orbit of ⟨ρ•⟩ corresponding to difference m 2 , and two edges from each orbit of ⟨ρ•⟩ cor￾responding to each difference d ∈ {2, 3, . . . , m−2 2 }, namely, a pair of edges of the form (e, ρ m 2 • (e)). The black orbits corr… view at source ↗
Figure 3
Figure 3. Figure 3: Lemma 7.2 – Starter m-cycles for an HOP (Cm)-decomposition of 4G• for m ≡ 2 (mod 4). Lemma 7.3 Let m ≡ 2 (mod 4), and let n be an odd integer such that 6 ≤ m < n. If n(n − 1) ≡ m (mod 2m), then 4K• n admits an HOP (Cm)-decomposition. Proof. By Lemma 6.6, it is enough to prove the result for odd n satisfying m < n < 2m. Write n = m+r+1, where 0 ≤ r < m−1 is even. If r = 0, then n = m+1, and by Lemma 6.5, th… view at source ↗
Figure 4
Figure 4. Figure 4: Lemma 8.1, Subcase 2.3 – The starter peripheral cycle C0 and the four starter central cycles for m = 8 and n = 13. peripheral cycles. We begin by outlining the parameters and then explain how many starter central and starter peripheral cycles are needed. Let V (4K• n ) = {xi : i ∈ Zn−1} ∪ {x∞}, and let ρ• be the permutation on E(4K• n ) that preserves the color (and orientation) of the edges, and is induce… view at source ↗
Figure 5
Figure 5. Figure 5: Orientation and coloring of the edges of [PITH_FULL_IMAGE:figures/full_fig_p048_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of the path Pf for ℓ ≡ 3 (mod 4). (i) ℓ−3 4 < 2 da ′ − ℓ+1 4 . This is proved in (39). (ii) ℓ − 2 d−1a ′ > 3 · 2 d−1a ′ − ℓ+1 2 , which is equivalent to 3ℓ+1 2 > 2 d+1a ′ . The latter follows from (38). Therefore, the internal vertices of the path Pf are pairwise distinct modulo ℓ. 53 [PITH_FULL_IMAGE:figures/full_fig_p053_6.png] view at source ↗
read the original abstract

The generalized honeymoon Oberwolfach problem (HOP) asks whether it is possible to seat $2n$ participants consisting of $n$ newlywed couples at a conference with $s$ tables of size $2$ and $t$ "round'' tables of sizes $2m_1, 2m_2, \ldots, 2m_t$, where $n = s + \sum_{i=1}^{t} m_i $ with all $m_i \geq 2$, over several nights so that each participant sits next to their spouse every time and next to each other participant exactly once. We denote this problem by $HOP(2^{\langle s \rangle}, 2m_1, \ldots, 2m_t)$. In this paper, we provide a complete solution to the generalized HOP with one round table, showing that the obvious necessary conditions for $HOP(2^{\langle s \rangle}, 2m)$ to have a solution are also sufficient.

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 / 1 minor

Summary. The paper claims to completely solve the generalized honeymoon Oberwolfach problem HOP(2^{<s>}, 2m) by proving that the standard divisibility conditions on the parameters n, s, and m are necessary and sufficient for the existence of the required seating arrangements over multiple nights, where each participant sits next to their spouse every night and next to every other participant exactly once. The solution is achieved via explicit constructions and exhaustive case analysis covering the full parameter range.

Significance. If the constructions and case analysis hold, the result provides a full existence theorem for the one-round-table case of the generalized HOP, closing a long-standing question in resolvable graph decompositions and Oberwolfach-type problems. The explicit constructions constitute a concrete, verifiable contribution that strengthens the sufficiency claim beyond mere existence arguments.

minor comments (1)
  1. The notation 2^{<s>} in the problem statement could be clarified with an explicit definition or reference to prior literature on the first occurrence in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, for confirming that the explicit constructions and case analysis address the full parameter range, and for recommending acceptance. The report accurately captures the contribution as a complete existence theorem for the one-round-table case.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper is a pure existence theorem in combinatorial design theory. It establishes sufficiency of the standard divisibility conditions for HOP(2^{<s>}, 2m) via explicit constructions and exhaustive case analysis on the parameters. No equations appear that equate a derived quantity to a fitted input, no predictions reduce to the input data by construction, and the central claim does not rest on a self-citation chain or imported uniqueness theorem from the author's prior work. The derivation is self-contained as a direct proof of the stated theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard combinatorial modeling of seating as graph decompositions; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Standard axioms of graph theory and block designs used to model the seating constraints
    The HOP is formulated as a decomposition problem whose basic properties are taken from prior literature.

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Reference graph

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15 extracted references · 1 canonical work pages

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