Long Monotone Trails in Random Edge-Labellings of Random Graphs
Bibliographic record
Abstract
Abstract Given a graph G and a bijection f : E ( G ) → {1, 2,…, e ( G )}, we say that a trail/path in G is f - increasing if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing worst-case estimates of the length of the longest increasing trail/path over all edge orderings of K n . The case of a trail was resolved by Graham and Kleitman, who proved that the answer is n -1, and the case of a path is still wide open. Recently Lavrov and Loh proposed studying the average-case version of this problem, in which the edge ordering is chosen uniformly at random. They conjectured (and Martinsson later proved) that such an ordering with high probability (w.h.p.) contains an increasing Hamilton path. In this paper we consider the random graph G = G n,p with an edge ordering chosen uniformly at random. In this setting we determine w.h.p. the asymptotics of the number of edges in the longest increasing trail. In particular we prove an average-case version of the result of Graham and Kleitman, showing that the random edge ordering of K n has w.h.p. an increasing trail of length (1- o (1)) en , and that this is tight. We also obtain an asymptotically tight result for the length of the longest increasing path for random Erdős-Renyi graphs with p = o (1).
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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 | 0.003 | 0.026 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".