Bibliographic record
Abstract
In this thesis we consider two variants on graph colouring.The őrst variant, ℓ-vertex-ranking requires that the vertices in the graph are assigned integer colours such that any path of length at most ℓ has a unique maximum colour.For this problem we give asymptotically tight bounds on the number of colours required for ℓ-vertex-ranking of planar graphs, solving a problem of Karpas, Neiman, and Smorodinsky (Discrete Mathematics, 2015).One of the tools used to establish the preceding results is layered partitions that appear in the context of graph product structure theory.In order to understand the limits of this approach we consider the relationship between layered partitions and the earlier notion of layered treewidth.We show that these two notions are strongly separated, so graphs admitting layered partitions are considerably more restricted than graphs with small layered treewidth.This work helps to explain why so much recent progress has been made using layered partitions on problems that resisted attack using layered treewidthThe second graph colouring variant we consider is linear colouring, which requires that we colour the vertices of a graph so that any path (regardless of length) has a vertex whose colour is unique.For this problem we consider the relationship between the minimum number of colours used in a linear-colouring (the linear chromatic number) and the treedepth of the graph.We give tighter upper bounds on the treedepth in terms of its linear chromatic number.This improves results of Czerwinski, Nadara, and Pilipczuk (SIAM J. Disc.Math., 2021) and Kun, O'Brien, Pilipczuk, and Sullivan (Algorithmica, 2021).It also gives further evidence for a bold conjecture of Kun et al on the relationship between linear colouring and treedepth.
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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 | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.009 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.014 | 0.003 |
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".