The analysis of a list-coloring algorithm on a random graph
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Bibliographic record
Abstract
We introduce a natural k-coloring algorithm and analyze its performance on random graphs with constant expected degree c (G/sub n,p=c/n/). For k=3 our results imply that almost all graphs with n vertices and 1.923 n edges are 3-colorable. This improves the lower bound on the threshold for random 3-colorability significantly and settles the last case of a long-standing open question of Bollobas. We also provide a tight asymptotic analysis of the algorithm. We show that for all k/spl ges/3, if c/spl les/k In k-3/2k then the algorithm almost surely succeeds, while for any /spl epsiv/>0, and k sufficiently large, if c/spl ges/(1+/spl epsiv/)k In k then the algorithm almost surely fails. The analysis is based on the use of differential equations to approximate the mean path of certain Markov chains.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 it