Bibliographic record
Abstract
Abstract For a pair of integers k , l ≥0, a graph G is ( k, l )‐colorable if its vertices can be partitioned into at most k independent sets and at most l cliques. The bichromatic number χ b ( G ) of G is the least integer r such that for all k , l with k + l = r , G is ( k, l )‐colorable. The concept of bichromatic numbers simultaneously generalizes the chromatic number χ( G ) and the clique covering number θ( G ), and is important in studying the speed of hereditary properties and edit distances of graphs. It is easy to see that for every graph G the bichromatic number χ b ( G ) is bounded above by χ( G )+θ( G )−1. In this article, we characterize all graphs G for which the upper bound is attained, i.e., χ b ( G )=χ( G )+θ( G )−1. It turns out that all these graphs are cographs and in fact they are the critical graphs with respect to the ( k, l )‐colorability of cographs. More specifically, we show that a cograph H is not ( k, l )‐colorable if and only if H contains an induced subgraph G with χ( G )= k +1, θ( G )= l +1 and χ b ( G )= k + l +1. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 263–269, 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 | 0.003 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| 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".