Bibliographic record
Abstract
Abstract Given a “forbidden graph” F and an integer k , an F‐avoiding k‐coloring of a graph G is a k ‐coloring of the vertices of G such that no maximal F ‐free subgraph of G is monochromatic. The F‐avoiding chromatic number ac F ( G ) is the smallest integer k such that G is F ‐avoiding k ‐colorable. In this paper, we will give a complete answer to the following question: for which graph F , does there exist a constant C , depending only on F , such that ac F ( G ) ⩽ C for any graph G ? For those graphs F with unbounded avoiding chromatic number, upper bounds for ac F ( G ) in terms of various invariants of G are also given. Particularly, we prove that \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${{ac}}_{{{F}}}({{G}})\le {{2}}\lceil\sqrt{{{n}}}\rceil+{{1}}$\end{document} , where n is the order of G and F is not K k or \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$\overline{{{K}}_{{{k}}}}$\end{document} . © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 300–310, 2010
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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 | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".