Bibliographic record
Abstract
One of the fundamental properties of the hypercube \( Q_n \) is that it is bipancyclic as \( Q_n \) has a cycle of length \( l \) for every even integer \( l \) with \( 4 \leq l \leq 2^n \). We consider the following problem of generalizing this property: For a given integer \( k \) with \( 3 \leq k \leq n \), determine all integers \( l \) for which there exists an \( l \)-vertex, \( k \)-regular subgraph of \( Q_n \) that is both \( k \)-connected and bipancyclic. The solution to this problem is known for \( k = 3 \) and \( k = 4 \). In this paper, we solve the problem for \( k = 5 \). We prove that \( Q_n \) contains a \( 5 \)-regular subgraph on \( l \) vertices that is both \( 5 \)-connected and bipancyclic if and only if \( l \in \{32, 48\} \) or \( l \) is an even integer satisfying \( 52 \leq l \leq 2^n \). For general \( k \), we establish that every \( k \)-regular subgraph of \( Q_n \) has \( 2^k, 2^k + 2^{k-1} \) or at least \( 2^k + 2^{k-1} + 2^{k-3} \) vertices.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".