A solution to small cases of the honeymoon Oberwolfach problem
Bibliographic record
Abstract
<p>The honeymoon Oberwolfach problem HOP<span class="math inline">\((2m_1,2m_2,\ldots,2m_t)\)</span> asks the following question. Given <span class="math inline">\(n=m_1+m_2+\ldots +m_t\)</span> newlywed couples at a conference and <span class="math inline">\(t\)</span> round tables of sizes <span class="math inline">\(2m_1,2m_2,\ldots,2m_t\)</span>, is it possible to arrange the <span class="math inline">\(2n\)</span> participants at these tables for <span class="math inline">\(2n-2\)</span> meals so that each participant sits next to their spouse at every meal, and sits next to every other participant exactly once? A solution to HOP<span class="math inline">\((2m_1,2m_2,\ldots,2m_t)\)</span> is a decomposition of <span class="math inline">\(K_{2n}+(2n-3)I\)</span>, the complete graph <span class="math inline">\(K_{2n}\)</span> with <span class="math inline">\(2n-3\)</span> additional copies of a fixed 1-factor <span class="math inline">\(I\)</span>, into 2-factors, each consisting of disjoint <span class="math inline">\(I\)</span>-alternating cycles of lengths <span class="math inline">\(2m_1,2m_2,\ldots,2m_t\)</span>. The honeymoon Oberwolfach problem was introduced in a 2019 paper by Lepine and Šajna. The authors conjectured that HOP<span class="math inline">\((2m_1,2m_2,\ldots,\)</span> <span class="math inline">\(2m_t)\)</span> has a solution whenever the obvious necessary conditions are satisfied, and proved the conjecture for several large cases, including the uniform cycle length case <span class="math inline">\(m_1=\ldots=m_t\)</span>, and the small cases with <span class="math inline">\(n \le 9\)</span>. In the present paper, we extend the latter result to all cases with <span class="math inline">\(n \le 20\)</span> using a computer-assisted search.</p>
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".