Decomposing hypercubes into cycles: An approach to the oberwolfach problem
Bibliographic record
Abstract
<p>Cartesian-product networks combine well-studied graphs to create new structures with inherited properties, making them valuable for interconnection networks and parallel algorithms. Cycle decompositions of these networks are crucial for fault tolerance and adaptive routing. In this paper, we address the hypercube version of the Oberwolfach problem, focusing on decompositions of <span class="math inline">\(Q_n\)</span> into cycles of equal or unequal lengths. We present an algorithm that enumerates all possible cycle types in <span class="math inline">\(Q_n\)</span> and determine which decompositions are feasible or infeasible for <span class="math inline">\(Q_4\)</span>. Using an inductive approach, we extend these results to <span class="math inline">\(Q_n\)</span> by leveraging distinct perfect matchings of <span class="math inline">\(Q_4\)</span>, yielding a variety of cycle decompositions. Additionally, we present results on factorizations of <span class="math inline">\(Q_n\)</span> when <span class="math inline">\(n\)</span> is a power of <span class="math inline">\(2\)</span>. These findings enhance the understanding of cycle structures in hypercubes and their applications to interconnection networks.</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 | 0.008 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.003 | 0.001 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.002 |
| 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".