Characterizing circular colouring mixing for pq<4 $\frac{p}{q}\lt 4$
Bibliographic record
Abstract
Abstract Given a graph , the ‐mixing problem asks: Starting with a ‐colouring of , can one obtain all ‐colourings of by changing the colour of only one vertex at a time, while at each step maintaining a ‐colouring? More generally, for a graph , the ‐mixing problem asks: Can one obtain all homomorphisms , starting from one homomorphism , by changing the image of only one vertex at a time, while at each step maintaining a homomorphism ? This paper focuses on a generalization of ‐colourings, namely, ‐circular colourings. We show that when , a graph is ‐mixing if and only if for any ‐colouring of , and any cycle of , the wind of the cycle under the colouring equals a particular value (which intuitively corresponds to having no wind). As a consequence we show that ‐mixing is closed under a restricted homomorphism called a fold. Using this, we deduce that ‐mixing is co‐NP‐complete for all , and by similar ideas we show that if the circular chromatic number of a connected graph is , then folds to . We use the characterization to settle a conjecture of Brewster and Noel, specifically that the circular mixing number of bipartite graphs is 2. Lastly, we give a polynomial time algorithm for ‐mixing in planar graphs when .
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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 | 0.007 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| 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".