Uniquely <i>D</i>-colourable Digraphs with Large Girth
Bibliographic record
Abstract
Abstract Let C and D be digraphs. A mapping f : V ( D ) → V ( C ) is a C -colouring if for every arc uv of D , either f ( u ) f ( v ) is an arc of C or f ( u ) = f ( v ), and the preimage of every vertex of C induces an acyclic subdigraph in D . We say that D is C -colourable if it admits a C -colouring and that D is uniquely C -colourable if it is surjectively C -colourable and any two C -colourings of D differ by an automorphism of C . We prove that if a digraph D is not C -colourable, then there exist digraphs of arbitrarily large girth that are D -colourable but not C -colourable. Moreover, for every digraph D that is uniquely D -colourable, there exists a uniquely D -colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number r ≥ 1, there are uniquely circularly r -colourable digraphs with arbitrarily large girth.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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".