Vizing’s Theorem in Near-Linear Time
Bibliographic record
Abstract
Vizing’s theorem states that any n-vertex m-edge graph of maximum degree Δ can be edge colored using at most Δ + 1 different colors [Vizing, 1964]. Vizing’s original proof is algorithmic and shows that such an edge coloring can be found in O(mn) time. This was subsequently improved to Õ(m√n) time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. Very recently, independently and concurrently, using randomization, this runtime bound was further improved to Õ(n2) by [Assadi, 2024] and Õ(mn1/3) by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to Õ(mn1/4) by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this paper, we present a randomized algorithm that computes a (Δ+1)-edge coloring in near-linear time—in fact, only O(mlogΔ) time—with high probability, giving a near-optimal algorithm for this fundamental problem.
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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 | 0.004 | 0.020 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.004 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.005 | 0.010 |
| Open science | 0.006 | 0.006 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.015 | 0.009 |
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".