A catalog of facially complete graphs
Bibliographic record
Abstract
onsidering regions in a map to be adjacent when they have nonempty intersection (as opposed to the traditional view requiring intersection in a linear segment) leads to the concept of a facially complete graph: a plane graph that becomes complete when edges are added between every two vertices that lie on a face. Here we present a complete catalog of facially complete graphs: they fall into seven types. A consequence is that if \(q\) is the size of the largest face in a plane graph \(G\) that is facially complete, then \(G\) has at most \(\left\lfloor 3q/2\right\rfloor\) vertices. This bound was known, but our proof is completely different from the 1998 approach of Chen, Grigni, and Papadimitriou (Planar map graphs, Proc. 30th ACM Symp. Th. of Computing, 514–523). Our method also yields a count of the 2-connected facially complete graphs with \(n\) vertices. We also show that if a plane graph has at most two faces of size 4 and no larger face, then the addition of both diagonals to each 4-face leads to a graph that is 5-colorable.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.005 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.015 | 0.002 |
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".