Bibliographic record
Abstract
We introduce a parameter called the level of nonmultiplicativity of a graph, which is related to Hedetniemi's conjecture. We show that this parameter is equal to the number of factors in a factorization of the graph into a product of multiplicative graphs. Apart from the known multiplicative graphs, no graph is known to have a nite level of nonmultiplicativity. We show that the countably innite complete graph K @0 has an innite level of nonmultiplicativity and that there exist Kneser graphs with arbitrarily high levels of nonmultiplicativity. 1 Introduction Given graphs G and H, the categorical product GH of G and H has vertex set V (GH) = f(g; h) : g 2 V (G) and h 2 V (H)g and edge set E(GH) = Supported in part by the National Science Council of R. O. C. under grant NSC882115 -M-110-001. 1 f(g; h)(g 0 ; h 0 ) : gg 0 2 E(G) and hh 0 2 E(H)g. This product is also called the tensor product. If c is an n-coloring of G, then it is straightforward to verify that ...
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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 | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".