Degree criteria and stability for independent transversals
Bibliographic record
Abstract
Abstract An independent transversal (IT) in a graph with a given vertex partition is an independent set of vertices of (i.e., it induces no edges), that consists of one vertex from each part ( block ) of . Over the years, various criteria have been established that guarantee the existence of an IT, often given in terms of being ‐ thick , meaning all blocks have size at least . One such result, obtained recently by Wanless and Wood, is based on the maximum average block degree . They proved that if then an IT exists. Resolving a problem posed by Groenland, Kaiser, Treffers and Wales (who showed that the ratio 1/4 is best possible), here we give a full characterization of pairs such that the following holds for every : whenever is a graph with maximum degree , and is a ‐thick vertex partition of such that , there exists an IT of with respect to . Our proof makes use of another previously known criterion for the existence of ITs that involve the topological connectedness of the independence complex of graphs, and establishes a general technical theorem on the structure of graphs for which this parameter is bounded above by a known quantity. Our result interpolates between the criterion and the old and frequently applied theorem that if then an IT exists. Using the same approach, we also extend a theorem of Aharoni, Holzman, Howard and Sprüssel, by giving a stability version of the latter result.
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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 | 0.003 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".