Bibliographic record
Abstract
This thesis concerns the relationship between four graph invariants: omega, chi_f, chi, and Delta. These are the clique number, the fractional chromatic number, the chromatic number, and the maximum degree, respectively. Trivially omega <= chi_f <= chi <= Delta + 1. We seek to improve the upper bound on chi. We are motivated by a conjecture of Reed, which essentially states that chi is at most the average of its trivial upper and lower bounds: Conjecture. For any graph, chi <= (Delta + 2 + omega)/2. We call this the Main Conjecture, and propose a Local Strengthening based on the closed neighbourhood of a single vertex: Conjecture. For any graph G, chi <= max{v in V(G)} (d(v) + 2 + omega(G[N(v)]) + 1) / 2. We begin by showing that much of the early evidence supporting the Main Conjecture also supports the Local Strengthening. In particular, the variant of the Local Strengthening obtained by replacing chi by chi_f holds, as does the Local Strengthening when the stability number is two. Guided by the first of these results we look towards line graphs, for which chi_f and chi agree asymptotically. We prove the Main Conjecture for line graphs, then we seek to generalize this result. To do this we use recent results of Chudnovsky and Seymour, who characterized the structure of all claw-free graphs. We refine their results by introducing a graph reduction on certain types of homogeneous pairs of cliques that preserves the chromatic number. Thus we need only consider the problem of colouring _skeletal_ claw-free graphs, which cannot be reduced. The structure of skeletal claw-free graphs is simpler than that of general claw-free graphs. We generalize two results from line graphs to the class of quasi-line graphs. Namely, that the Main Conjecture holds, and that chi_f and chi agree asymptotically. We then consider all claw-free graphs. We prove the Main Conjecture for all claw-free graphs and we prove the Local
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| 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".