Hamilton Cycles in Restricted and Incomplete Rotator Graphs
Bibliographic record
Abstract
The nodes of a rotator graph are the permutations of n, and an arc is directed from u to v if the first r symbols of u can be rotated one position to the left to obtain v. Restricted rotator graphs restrict the allowable rotations to r ∈ R for some R ⊆ {2,3,…,n}. Incomplete rotator graphs only include nodes whose final symbol is i ≤ m for a fixed maximum value m ∈ {1,2,…,n}. Restricted rotator graphs are directed Cayley graphs, whereas incomplete rotator graphs are not Cayley graphs. Hamilton cycles exist for rotator graphs (Corbett 1992), restricted rotator graphs with R={n−1,n} (Ruskey and Williams 2010), and incomplete rotator graphs for all m (Ponnuswamy and Chaudhary 1994). These previous results are based on sequence building operations that we name `reusing', `recycling', and `rewinding'. In this article, we combine these operations to create Hamilton cycles in rotator graphs that are (1) restricted by R={2,3,n}, (2) restricted by R={2,3,n−1,n} and incomplete for any m, and (3) restricted by R={n−2,n−1,n} and incomplete for any m. Result (1) is `optimal' since restricted rotator graphs are not strongly connected for R={3,n} when n is odd, and do not have Hamilton cycles for R={2,n} when n is even (Rankin 1944, Swan 1999). Similarly, we prove (3) is `optimal'. Our Hamilton cycles can be easily implemented for potential applications, and we provide O(1)-time algorithms that generate successive rotations for (1)-(3).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".