On the oriented chromatic number of dense graphs
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Bibliographic record
Abstract
Let G be a graph with n vertices, m edges, average degree δ, and maximum degree Δ. The \emph{oriented chromatic number} of G is the maximum, taken over all orientations of G, of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which δ≥logn. We prove that every such graph has oriented chromatic number at least Ω(√n). In the case that δ≥(2+ϵ)logn, this lower bound is improved to Ω(√m). Through a simple connection with harmonious colourings, we prove a general upper bound of \OhΔ√n on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when G is (clogn)-regular for some constant c>2, in which case the oriented chromatic number is between Ω(√nlogn) and O(√nlogn).
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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 it