Flexible list colorings in graphs with special degeneracy conditions
Bibliographic record
Abstract
Abstract For a given , we say that a graph is ‐ flexibly ‐ choosable if the following holds: for any assignment of color lists of size on , if a preferred color from a list is requested at any set of vertices, then at least of these requests are satisfied by some ‐coloring. We consider the question of flexible choosability in several graph classes with certain degeneracy conditions. We characterize the graphs of maximum degree that are ‐flexibly ‐choosable for some , which answers a question of Dvořák Norin, and Postle [List coloring with requests, JGT 2019]. In particular, we show that for any , any graph of maximum degree that is not isomorphic to is ‐flexibly ‐choosable. Our fraction of is within a constant factor of being the best possible. We also show that graphs of treewidth 2 are ‐flexibly 3‐choosable, answering a question of Choi et al. [Flexibility of planar graphs‐sharpening the tools to get lists of size four, DAM 2022], and we give conditions for list assignments by which graphs of treewidth are ‐flexibly ‐choosable. We show furthermore that graphs of treedepth are ‐flexibly ‐choosable. Finally, we introduce a notion of flexible degeneracy , which strengthens flexible choosability, and we show that apart from a well‐understood class of exceptions, 3‐connected nonregular graphs of maximum degree are flexibly ‐degenerate.
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 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".