A complete solution to the spectrum problem for graphs with six vertices and up to nine edges
Bibliographic record
Abstract
Let $G$ be a graph. A $G$-design of order $n$ is a decomposition of the complete graph $K_n$ into disjoint copies of $G$. The existence problem of graph designs has been completely solved for all graphs with up to five vertices, and all graphs with six vertices and up to seven edges; and almost completely solved for all graphs with six vertices and eight edges leaving two cases of order 32 unsettled. Up to isomorphism there are 20 graphs with six vertices and nine edges (and no isolated vertex). The spectrum problem has been solved completely for 11 of these graphs, and partially for 2 of these graphs. In this article, the two missing graph designs for the six-vertex eight-edge graphs are constructed, and a complete solution to the spectrum problem for the six-vertex nine-edge graphs is given; completing the spectrum problem for all graphs with six vertices and up to nine edges.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.018 | 0.003 |
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".