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 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 | 0.003 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.005 | 0.008 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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".