List-3-Coloring Ordered Graphs with a Forbidden Induced Subgraph
Bibliographic record
Abstract
Abstract. The List-3-Coloring Problem is to decide, given a graph [Formula: see text] and a list [Formula: see text] of colors assigned to each vertex [Formula: see text] of [Formula: see text], whether [Formula: see text] admits a proper coloring [Formula: see text] with [Formula: see text] for every vertex [Formula: see text] of [Formula: see text], and the 3-Coloring Problem is the List-3-Coloring Problem on instances with [Formula: see text] for every vertex [Formula: see text] of [Formula: see text]. The List-3-Coloring Problem is a classical NP -complete problem, and it is well-known that while restricted to [Formula: see text]- free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph [Formula: see text]), it remains NP -complete unless [Formula: see text] is isomorphic to an induced subgraph of a path. However, the current state of art is far from proving this to be sufficient for a polynomial time algorithm; in fact, the complexity of the 3-Coloring Problem on [Formula: see text]-free graphs (where [Formula: see text] denotes the eight-vertex path) is unknown. Here we consider a variant of the List-3-Coloring Problem called the Ordered Graph List-3-Coloring Problem, where the input is an ordered graph, that is, a graph along with a linear order on its vertex set. For ordered graphs [Formula: see text] and [Formula: see text], we say [Formula: see text] is [Formula: see text]- free if [Formula: see text] is not isomorphic to an induced subgraph of [Formula: see text] with the isomorphism preserving the linear order. We prove, assuming [Formula: see text] to be an ordered graph, a nearly complete dichotomy for the Ordered Graph List-3-Coloring Problem restricted to [Formula: see text]-free ordered graphs. In particular, we show that the problem can be solved in polynomial time if [Formula: see text] has at most one edge, and remains NP -complete if [Formula: see text] has at least three edges. Moreover, in the case where [Formula: see text] has exactly two edges, we give a complete dichotomy when the two edges of [Formula: see text] share an end, and prove several NP -completeness results when the two edges of [Formula: see text] do not share an end, narrowing the open cases down to three very special types of two-edge ordered graphs.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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".