Cycles in 3‐connected claw‐free planar graphs and 4‐connected planar graphs without 4‐cycles
Bibliographic record
Abstract
Abstract The cycle spectrum of a graph is the set of the cycle lengths in . Let be a graph class. For any integer , define to be the least integer such that for any with circumference at least . Denote by , and the classes of 3‐connected planar graphs, 3‐connected cubic planar graphs, 3‐connected claw‐free planar graphs, and 3‐connected claw‐free cubic planar graphs, respectively. The values of and were known for all . In the first part of this article, we prove the claw‐free version of these results by giving the values of and for all . In the second part we study the cycle spectra of 4‐connected planar graphs without 4‐cycles. Bondy conjectured that every 4‐connected planar graph has all cycle lengths from 3 to except possibly one even length. The truth of this conjecture would imply that every 4‐connected planar graph without 4‐cycles has all cycle lengths other than 4. It was already known that . We prove that can be embedded in the plane such that for any has a ‐cycle and all vertices not in lie in the exterior of .
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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 | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.002 | 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".