Overlap number of graphs
Bibliographic record
Abstract
Abstract An overlap representation of a graph G assigns sets to vertices so that vertices are adjacent if and only if their assigned sets intersect with neither containing the other. The overlap number φ( G ) (introduced by Rosgen) is the minimum size of the union of the sets in such a representation. We prove the following: (1) An optimal overlap representation of a tree can be produced in linear time, and its size is the number of vertices in the largest subtree in which the neighbor of any leaf has degree 2. (2) If δ( G )⩾2 and G ≠ K 3 , then φ( G )⩽| E ( G )| − 1, with equality when G is connected and triangle‐free and has no star‐cutset. (3) If G is an n ‐vertex plane graph with n ⩾5, then φ( G )⩽2 n − 5, with equality when every face has length 4 and there is no star‐cutset. (4) If G is an n ‐vertex graph with n ⩾14, then φ( G )⩽ n 2 /4 − n /2 − 1, with equality for even n when G arises from K n /2, n /2 by deleting a perfect matching. © 2011 Wiley Periodicals, Inc. J Graph Theory
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".