On Coloring a Class of Claw-free Graphs
Bibliographic record
Abstract
Given a set L of graphs, a graph G is L -free if G does not contain any graph in L as an induced subgraph. Recently, Frédéric Maffray and co-authors showed that the problem of coloring { claw , 4 K 1 , K 5 \ e }-free graphs can be solved in polynomial time. In this paper, we investigate a related class of graphs. A hole is an induced cycle of length at least 4. Two vertices x , y of a graph G are twins if for any vertex z different from x and y , xz is an edge if and only if yz is an edge. A hole-twin is the graph obtained from a hole by adding a vertex that forms a twin with some vertex of the hole. Hole-twins, and K 5 \ e , are interesting in their connection with line-graphs. They are among the forbidden subgraphs in the characterization of line-graphs. In this paper, we show there is a polynomial time algorithm to color ( claw , 4 K 1 , hole-twin)-free 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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 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".