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 machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".