The Complexity of two Colouring Games
Bibliographic record
Abstract
We consider two variants of orthogonal colouring games on graphs. In these games, two players alternate colouring uncoloured vertices (from a choice of $m\in \mathbb{N}$ colours) of a pair of isomorphic graphs while respecting the properness and the orthogonality of the partial colourings. In the normal play variant, the first player unable to move loses. In the scoring variant, each player aims to maximise their score, which is the number of coloured vertices in their copy of the graph. We prove that, given an instance with a partial colouring, both the normal play and the scoring variant of the game are PSPACE-complete. An involution $\sigma$ of a graph G is strictly matched if its fixed point set induces a clique and $v\sigma(v)\in E(G)$ for any non-fixed point $v\in V(G)$. Andres, Huggan, Mc Inerney, and Nowakowski (The orthogonal colouring game. Theor. Comput. Sci., 795:312-325, 2019) gave a solution of the normal play variant played on graphs that admit a strictly matched involution. We prove that recognising graphs that admit a strictly matched involution is NP-complete.
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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 | 0.011 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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".