Four-Coloring \(\boldsymbol{P_6}\)-Free Graphs. II. Finding an Excellent Precoloring
Bibliographic record
Abstract
This is the second paper in a series of two. The goal of the series is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time-algorithm that starts with a 4-precoloring of a graph with no induced six-vertex path and outputs a polynomial-sized collection of so-called excellent precolorings. Excellent precolorings are easier to handle than general ones, and, in addition, in order to determine whether the initial precoloring can be extended to the whole graph, it is enough to answer the same question for each of the excellent precolorings in the collection. The first paper in the series deals with excellent precolorings, thus providing a complete solution to the problem.
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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 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 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".