On the generalized Oberwolfach problem
Bibliographic record
Abstract
The generalized Oberwolfach problem OPt(2w + 1; N1, N2, …, Nt; α1, α2, …, αt) asks for a factorization of K2w + 1 into αi CNi-factors (where a CNi-factor of K2w + 1 is a spanning subgraph whose components are cycles of length Ni ≥ 3) for i = 1, 2, …, t. Necessarily, N = lcm(N1, N2, …, Nt) is a divisor of 2w + 1 and w = Σ ti = 1 αi.For t = 1 we have the classic Oberwolfach problem. For t = 2 this is the well-studied Hamilton-Waterloo problem, whereas for t ≥ 3 very little is known.In this paper, we show, among other things, that the above necessary conditions are sufficient whenever 2w + 1 ≥ (t + 1)N, αi > 1 for every i ∈ {1, 2, …, t}, and gcd (N1, N2, …, Nt) > 1. We also provide sufficient conditions for the solvability of the generalized Oberwolfach problem over an arbitrary graph and, in particular, the complete equipartite graph.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.013 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".