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 G and a list L(v) ⊆ \n{1, 2, 3} of colors assigned to each vertex v of G, whether G admits a proper coloring ϕ with \nϕ(v) ∈ L(v) for every vertex v of G, and the 3-Coloring Problem is the List-3-Coloring \nProblem on instances with L(v) = {1, 2, 3} for every vertex v of G. The List-3-Coloring \nProblem is a classical NP-complete problem, and it is well-known that while restricted to \nH-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph H), it \nremains NP-complete unless H is isomorphic to an induced subgraph of a path. However, the \ncurrent state of art is far from proving this to be sufficient for a polynomial time algorithm; \nin fact, the complexity of the 3-Coloring Problem on P8-free graphs (where P8 denotes the \neight-vertex path) is unknown. Here we consider a variant of the List-3-Coloring Problem \ncalled the Ordered Graph List-3-Coloring Problem, where the input is an ordered graph, \nthat is, a graph along with a linear order on its vertex set. For ordered graphs G and H, we \nsay G is H-free if H is not isomorphic to an induced subgraph of G with the isomorphism \npreserving the linear order. We prove, assuming H to be an ordered graph, a nearly complete \ndichotomy for the Ordered Graph List-3-Coloring Problem restricted to H-free ordered \ngraphs. In particular, we show that the problem can be solved in polynomial time if H has at \nmost one edge, and remains NP-complete if H has at least three edges. Moreover, in the case \nwhere H has exactly two edges, we give a complete dichotomy when the two edges of H share \nan end, and prove several NP-completeness results when the two edges of H do not share an \nend, 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 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 | 0.001 | 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 | 0.001 | 0.001 |
| Open science | 0.001 | 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".