Graph Colouring with No Large Monochromatic Components
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Bibliographic record
Abstract
For a graph G and an integer t we let mcc t ( G ) be the smallest m such that there exists a colouring of the vertices of G by t colours with no monochromatic connected subgraph having more than m vertices. Let be any non-trivial minor-closed family of graphs. We show that mcc 2 ( G ) = O ( n 2/3 ) for any n -vertex graph G ∈ . This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such , and every fixed t we show that mcc t ( G )= O ( n 2/( t +1) ). On the other hand, we have examples of graphs G with no K t +3 minor and with mcc t ( G )=Ω( n 2/(2 t −1) ). It is also interesting to consider graphs of bounded degrees. Haxell, Szabó and Tardos proved mcc 2 ( G ) ≤ 20000 for every graph G of maximum degree 5. We show that there are n -vertex 7-regular graphs G with mcc 2 ( G )=Ω( n ), and more sharply, for every ϵ > 0 there exists c ϵ > 0 and n -vertex graphs of maximum degree 7, average degree at most 6 + ϵ for all subgraphs, and with mcc 2 ( G ) ≥ c ϵ n . For 6-regular graphs it is known only that the maximum order of magnitude of mcc 2 is between $\sqrt n$ and n . We also offer a Ramsey-theoretic perspective of the quantity mcc t ( G ).
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 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 it