Strong 3‐Flow Conjecture for projective planar graphs
Bibliographic record
Abstract
Abstract In 1972, Tutte posed the 3‐Flow Conjecture: that all 4‐edge‐connected graphs have a nowhere‐zero 3‐flow. This was extended by Jaeger et al. to allow vertices to have a prescribed, possibly nonzero difference (modulo 3) between the inflow and outflow. They conjectured that all 5‐edge‐connected graphs with a prescription function have a nowhere‐zero 3‐flow meeting that prescription. Kochol showed that replacing 4‐edge‐connected with 5‐edge‐connected would suffice to prove the 3‐Flow Conjecture and Lovász et al. showed that both conjectures hold if the edge connectivity condition is relaxed to 6‐edge‐connected. Both problems are still open for 5‐edge‐connected graphs. The 3‐Flow Conjecture was known to hold for planar graphs, as it is the dual of Grötzsch's Colouring Theorem. Steinberg and Younger provided the first direct proof using flows for planar graphs, as well as a proof for projective planar graphs. Richter et al. provided the first direct proof using flows of the Strong 3‐Flow Conjecture for planar graphs. We prove the Strong 3‐Flow Conjecture for projective planar graphs.
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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 | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 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".