Bibliographic record
Abstract
A graph G is k -vertex-critical if χ ( G ) = k but χ ( G − v ) < k for all v ∈ V ( G ) and ( H 1 , H 2 ) -free if it contains no induced subgraph isomorphic to H 1 or H 2 . We show that there are only finitely many k -vertex-critical (co-gem, H )-free graphs for all k when H is any graph of order 4 by showing finiteness in the three remaining open cases, those are the cases when H is 2 P 2 , K 3 + P 1 , and K 4 . For the first two cases we actually prove the stronger results: • There are only finitely many k -vertex-critical (co-gem, paw + P 1 )-free graphs for all k ≥ 1 . • There are only finitely many k -vertex-critical (co-gem, P 5 , P 3 + c P 2 )-free graphs for all k ≥ 1 and c ≥ 0 . To prove the latter result, we employ a novel application of Sperner's Theorem on the number of antichains in a partially ordered set. Our result for K 4 uses exhaustive computer search and is proved by showing the stronger result that every ( co-gem, K 4 ) -free graph is 4-colourable. Our results imply the existence of simple polynomial-time certifying algorithms to decide the k -colourability of (co-gem, H )-free graphs for all k and all H of order 4 by searching the vertex-critical graphs as induced subgraphs.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.007 |
| Science and technology studies | 0.001 | 0.010 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.009 | 0.003 |
| 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".