Bibliographic record
Abstract
Many problems which seek to schedule, sequence, or time-table a set of events subject to given constraints can be modelled as graph colouring problems. In this thesis, we study the edge and total colouring problems which, like all NP problems, can be formulated as integer programs and subjected to a two-pronged attack: we first solve the fractional relaxation and then use this solution to solve or obtain an approximation of the solution of the integer program. We focus on the complexity of solving the fractional relaxations of the integer programs for the edge and total colouring problems. For each ε > 0, we give a linear time algorithm which determines the fractional chromatic index of a graph $G$ with maximum degree at least ε|G|. For graphs with large maximum degree, this improves on Padberg and Rao's polynomial time algorithm to determine the fractional chromatic index for general graphs. Both algorithms rely on a theorem of Edmonds showing that the fractional chromatic index of a graph is determined by its maximum degree and overfull subgraphs. Our algorithm exploits the fact that overfull subgraphs are related to small cuts in a graph and have simple intersection patterns when the maximum degree is large. The complexity of determining the fractional total colouring number is currently unresolved. We focus on graphs with large maximum degree, applying the very successful techniques for fractional edge colouring to fractional total colouring. We characterize graphs with maximum degree Δ whose fractional total colouring number is Δ + 2, sharpening a result of Kilakos and Reed who showed it is between Δ + 1 and Δ+2. We show graphs whose fractional total colouring number is less than Δ + 2 have a special fractional vertex colouring which extends to a fractional total colouring using less than Δ + 2 colours. We extend these ideas by giving necessary conditions a fractional vertex ß-colouring must satisfy to be extendable to a fractional total ß-colouring. We conjecture these conditions are sufficient when G satisfies Δ > ½|G|. We verify a special case of this conjecture by giving a polynomial time algorithm which constructs an optimal fractional total colouring of a graph G with maximum degree at least ¾|G| and containing no overfull subgraphs.
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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 | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 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".