Reconfiguration of colorings in triangulations of the sphere
Bibliographic record
Abstract
In 1973, Fisk proved that any $4$-coloring of a $3$-colorable triangulation of the $2$-sphere can be obtained from any $3$-coloring by a sequence of Kempe changes. On the other hand, in the case where we are only allowed to recolor a single vertex in each step, which is a special case of a Kempe change, there exists a $4$-coloring that cannot be obtained from any $3$-coloring. In this paper, we present a linear-time checkable characterization of a $4$-coloring of a $3$-colorable triangulation of the $2$-sphere that can be obtained from a $3$-coloring by a sequence of recoloring operations at single vertices. In addition, we develop a quadratic-time algorithm to find such a recoloring sequence if it exists; our proof implies that we can always obtain a quadratic length recoloring sequence. We also present a linear-time checkable criterion for a $3$-colorable triangulation of the $2$-sphere that all $4$-colorings can be obtained from a $3$-coloring by such a sequence. Moreover, we consider a high-dimensional setting. As a natural generalization of our first result, we obtain a polynomial-time checkable characterization of a $k$-coloring of a $(k-1)$-colorable triangulation of the $(k-2)$-sphere that can be obtained from a $(k-1)$-coloring by a sequence of recoloring operations at single vertices and the corresponding algorithmic result. Furthermore, we show that the problem of deciding whether, for given two $(k+1)$-colorings of a $(k-1)$-colorable triangulation of the $(k-2)$-sphere, one can be obtained from the other by such a sequence is PSPACE-complete for any fixed $k \geq 4$. Our results above can be rephrased as new results on the computational problems named $k$-Recoloring and Connectedness of $k$-Coloring Reconfiguration Graph, which are fundamental problems in the field of combinatorial reconfiguration.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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.
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".