Orientation‐based edge‐colorings and linear arboricity of multigraphs
Bibliographic record
Abstract
Abstract The Goldberg–Seymour Conjecture for ‐colorings states that the ‐chromatic index of a loopless multigraph is essentially determined by either a weighted maximum degree or a weighted maximum density parameter. We introduce an oriented version of ‐colorings, where now each color class of the edge‐coloring is required to be orientable in such a way that every vertex has indegree and outdegree at most some specified values and . We prove that the associated ‐oriented chromatic index satisfies a Goldberg–Seymour formula. We then present simple applications of this result to variations of ‐colorings. In particular, we show that the Linear Arboricity Conjecture holds for ‐degenerate loopless multigraphs when the maximum degree is at least , improving a recent bound by Chen, Hao, and Yu for simple graphs. Finally, we demonstrate that the ‐oriented chromatic index is always equal to its list coloring analogue.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".