Reconfiguring vertex colourings of 2-trees
Why this work is in the frame
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Bibliographic record
Abstract
Let H be a graph and let k ≥ χ(H) be an integer. The k-colouring graph of H, denoted Gk(H), is the graph whose vertex set consists of all proper k-vertex-colourings (or simply k-colourings) of H using colours {1, 2, …, k}; two vertices of Gk(H) are adjacent if and only if the corresponding k-colourings differ in colour on exactly one vertex of H. If Gk(H) has a Hamilton cycle, then H is said to have a Gray code of k-colourings, and the Gray code number of H is the least integer k0(H) such that Gk(H) has a Gray code of k-colourings for all k ≥ k0(H). Choo and MacGillivray determine the Gray code numbers of trees. We extend this result to 2-trees. A 2-tree is constructed recursively by starting with a complete graph on three vertices and connecting each new vertex to an existing clique on two vertices. We prove that if H is a 2-tree, then k0(H) = 4 unless H is isomorphic to the join of a tree T and a vertex u, where T is a star on at least three vertices, or the bipartition of T has two even parts; in these cases, k0(H) = 5.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| 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 it