Injective edge colorings of degenerate graphs and the oriented chromatic number
Bibliographic record
Abstract
Given a graph G , an injective edge-coloring of G is a function ψ : E ( G ) → N such that if ψ ( e ) = ψ ( e ′ ) , then no third edge joins an endpoint of e and an endpoint of e ′ . The injective chromatic index of a graph G , written χ inj ′ ( G ) , is the minimum number of colors needed for an injective edge coloring of G . In this paper, we investigate the injective chromatic index of certain classes of degenerate graphs. First, we show that if G is a d -degenerate graph of maximum degree Δ , then χ inj ′ ( G ) = O ( d 3 log Δ ) . Next, we show that if G is a graph of Euler genus g , then χ inj ′ ( G ) ≤ ( 3 + o ( 1 ) ) g , which is tight when G is a clique. Finally, we show that the oriented chromatic number of a graph is at most exponential in its injective chromatic index. Using this fact, we prove that the oriented chromatic number of a graph embedded on a surface of Euler genus g has oriented chromatic number at most O ( g 6400 ) , improving the previously known upper bound of 2 O ( g 1 2 + ɛ ) and resolving a conjecture of Aravind and Subramanian.
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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 | 0.002 | 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".