Hamilton Cycles in Restricted and Incomplete Rotator Graphs
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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 rst r symbols of u can be rotated one position to the left to obtain v. Restricted rotator graphs restrict the allowable rotations to r2 R for some Rf 2; 3;:::;ng. Incomplete rotator graphs only include nodes whose nal symbol is i m for a xed maximum value m2 f1; 2;:::;ng. 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 =fn 1;ng (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 =f2; 3;ng, (2) restricted by R = f2; 3;n 1;ng and incomplete for any m, and (3) restricted by R =fn 2;n 1;ng and incomplete for any m. Result (1) is ‘optimal’ since restricted rotator graphs are not strongly connected for R =f3;ng when n is odd, and do not have Hamilton cycles for R =f2;ng 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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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 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 | 0.000 | 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.
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