Saved by the rook: a case of matchings and Hamiltonian cycles
Bibliographic record
Abstract
The rook graph is a graph whose edges represent all the possible legal moves of the rook chess piece on a chessboard. The problem we consider is the following. Given any set $M$ containing pairs of cells such that each cell of the $m_1 \times m_2$ chessboard is in exactly one pair, we determine the values of the positive integers $m_1$ and $m_2$ for which it is possible to construct a closed tour of all the cells of the chessboard which uses all the pairs of cells in $M$ and some edges of the rook graph. This is an alternative formulation of a graph-theoretical problem presented in Marién Abreu, John Baptist Gauci, Domenico Labbate, Giuseppe Mazzuoccolo, and Jean Paul Zerafa, Extending perfect matchings to Hamiltonian cycles in line graphs, Electron. J. Comb. 28 (2021), no.~1, research paper p1.7, 13 (English) involving the Cartesian product $G$ of two complete graphs $K_{m_1}$ and $K_{m_2}$, which is, in fact, isomorphic to the $m_{1}\times m_{2}$ rook graph. The problem revolves around determining the values of the parameters $m_1$ and $m_2$ that would allow any perfect matching of the complete graph on the same vertex set of $G$ to be extended to a Hamiltonian cycle by using only edges in $G$.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.000 | 0.001 |
| 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".