Graph Colouring with No Large Monochromatic Components
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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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".