Bibliographic record
Abstract
For which values of k does a uniformly chosen 3‐regular graph G on n vertices typically contain n/k vertex‐disjoint k ‐cycles (a k ‐cycle factor)? To date, this has been answered for k = n and for k ≪ log n ; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability (w.h.p.) most vertices do not lie on k ‐cycles. A major role in our study of this problem is played by renewal processes without replacement, where one wishes to estimate the probability that in a uniform permutation of a given set of positive integers, the partial sums hit a designated target integer. Using sharp tail estimates for these renewal processes, which may be of independent interest, we settle the cycle factor problem completely: the “threshold” for a k ‐cycle factor in G as above is κ 0 log 2 n with . To be precise, G contains a k ‐cycle factor w.h.p. if urn:x-wiley:00103640:media:cpa21613:cpa21613-math-0002 and w.h.p. does not contain one if . Thus, for most values of n the threshold concentrates on the single integer K 0 ( n ). As a byproduct, we confirm the “comb conjecture,” an old problem concerning the embedding of certain spanning trees in the random graph ( n,p ).© 2015 Wiley Periodicals, Inc.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| 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".