Bibliographic record
Abstract
Abstract. A k-queue layout of a graph consists of a total order of the vertices, and a partition of the edges into k sets such that no two edges that are in the same set are nested with respect to the vertex ordering. A k-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The queue-number (track-number) of a graph G, is the minimum k such that G has a k-queue (k-track) layout. This paper proves that every n-vertex planar graph has track number and queue number at most O(logn). This improves the result of Di Battista, Frati and Pach [Founda-tions of Computer Science, (FOCS ’10), pp. 365–374] who proved the first sub-polynomial bounds on the queue number and track number of planar graphs. Specifically, they ob-tained O(log2n) queue number and O(log8n) track number bounds for planar graphs. The result also implies that every planar graph has a 3D crossing-free grid drawing in O(n logn) volume. The proof uses a non-standard type of graph separators. 1
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.005 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.018 | 0.004 |
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".