Bibliographic record
Abstract
A k-list-assignment for a graph G assigns to each vertex v of G a list L(v) ofadmissible colors, where|L(v) | ≥ k. A graph is k-list-colorable (or k-choosable) if it can be properly colored from the lists for every k-listassignment. We prove the following conjecture posed by Thomassen in 1994: “There are only finitely many listcolor-critical graphs with all lists of cardinality at least 5 on any fixed surface. ” This generalizes the well-known result of Thomassen on the usual graph coloring case. We use this theorem and specific parts of its proof to resolve the complexity status of the following problem about k-list-coloring graphs on a fixed surface S, where k is a fixed positive integer. Input: A graph G embedded in the surface S. Question: Is G k-choosable? If not, provide a certificate (a list-color-critical subgraph and the corresponding k-list-assignment). The cases k = 3, 4 are known to be NP-hard (actually even Π p 2-complete), and the cases k = 1, 2 are easy. Our main results imply that the problem is tractable for every k ≥ 5. In fact, together with our recent algorithmic result, we are able to solve it in linear time when k ≥ 5. Our proof yields even more: if the input graph is k-list-colorable, then for any k-listassignment L, we can construct an L-coloring of G in linear time. This generalizes the well-known linear-time algorithms for planar graphs by Nishizeki and Chiba (for 5-coloring), and Thomassen (for 5-list-coloring). We also give a polynomial-time algorithm to resolve the following question: Input: A graph G in the surface S, andak-listassignment L, wherek≥5.
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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 | 0.000 | 0.000 |
| 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 | 0.001 | 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 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".