Reconfiguration of Hamiltonian Cycles and Paths in Rectangular Grid Graphs
Bibliographic record
Abstract
An m×n grid graph is the induced subgraph of the square lattice whose vertex set consists of all integer grid points {(i, j) : 0 ≤ i < m, 0 ≤ j < n}. Let H and K be Hamiltonian cycles in an m × n grid graph G. We study the problem of reconfiguring H into K using a sequence of local transformations called moves. A box of G is a unit square face. A box with vertices a, b, c, d is switchable in H if exactly two of its edges belong to H, and these edges are parallel. Given such a box with edges ab and cd in H, a switch move removes ab and cd, and adds bc and ad. A double-switch move consists of performing two consecutive switch moves. If, after a double-switch move, we obtain a Hamiltonian cycle, we say that the double-switch move is valid. We prove that any Hamiltonian cycle H can be transformed into any other Hamiltonian cycle K via a sequence of valid double-switch moves, such that every intermediate graph remains a Hamiltonian cycle. This result extends to Hamiltonian paths. In that case, we also use single-switch moves and a third operation, the backbite move, which enables the relocation of the path endpoints.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".