Complements of nearly perfect graphs
Bibliographic record
Abstract
A class of graphs closed under taking induced subgraphs is χbounded if there exists a function f such that for all graphs G in the class, χ(G) ≤ f (ω(G)).We consider the following question initially studied in [A.Gyárfás, Problems from the world surrounding perfect graphs, Zastowania Matematyki Applicationes Mathematicae, 19:413-441, 1987].For a χ-bounded class C, is the class C χ-bounded (where C is the class of graphs formed by the complements of graphs from C)?We show that if C is χ-bounded by the constant function f (x) = 3, then C is χ-bounded by g(x) = 8 5 x and this is best possible.We show that for every constant c > 0, if C is χ-bounded by a function f such that f (x) = x for x ≥ c, then C is χ-bounded.For every j, we construct a class of graphs χ-bounded by f (x) = x + x/ log j (x) whose complement is not χ-bounded.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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 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".