Bibliographic record
Abstract
Abstract A classical theorem of Brooks in graph coloring theory states that every connected graph G has its chromatic number less than or equal to its maximum degree , unless G is a complete graph or an odd cycle in which case is equal to . Brooks' theorem has been extended to list colorings by Erdős, Rubin, and Taylor (and, independently, by Vizing) and to some of their variants such as list T ‐colorings and pair‐list colorings. The bichromatic number is a relatively new parameter arisen in the study of extremal hereditary properties of graphs. This parameter simultaneously generalizes the chromatic number and the clique covering number of a graph. In this article, we prove a theorem, akin to that of Brooks, which states that every graph G has its bichromatic number less than or equal to its bidegree , unless G belongs to a set of specified graphs in which case is equal to .
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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 | 0.007 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 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 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".