Oriented colouring graphs of bounded degree and degeneracy
Bibliographic record
Abstract
This paper considers upper bounds on the oriented chromatic number χ o ( G ) , of an oriented graph G in terms of its 2-dipath chromatic number χ 2 ( G ) , degeneracy d ( G ) , and maximum degree Δ ( G ) . In particular, we show that for all graphs G with χ 2 ( G ) ≤ k where k ≥ 2 and d ( G ) ≤ t where t ≥ log 2 ( k ) , χ o ( G ) = 33 / 10 ( k t 2 2 t ) . This improves an upper bound of MacGillivray, Raspaud, and Swartz of the form χ o ( G ) ≤ 2 χ 2 ( G ) − 1 to a polynomial upper bound for many classes of graphs, in particular, those with bounded degeneracy. Additionally, we asymptotically improve bounds for the oriented chromatic number in terms of maximum degree and degeneracy. For instance, we show that χ o ( G ) ≤ ( 2 ln 2 + o ( 1 ) ) Δ 2 2 Δ for all graphs, and χ o ( G ) ≤ ( 2 + o ( 1 ) ) Δ d 2 d for graphs where degeneracy grows sublinearly in maximum degree. Here the asymptotics are in Δ. The former improves the asymptotics of a results by Kostochka, Sopena, and Zhu [9] , while the latter improves the asymptotics of a result by Aravind and Subramanian [1] . Both improvements are by a constant factor.
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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 | 0.000 | 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 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".