Small cycle double covers of products I: Lexicographic product with paths and cycles
Why this work is in the frame
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Bibliographic record
Abstract
Abstract Bondy conjectured that every simple bridgeless graph has a small cycle double cover (SCDC). We show that this is the case for the lexicographic products of certain graphs and along the way for the Cartesian product as well. Specifically, if G does not have an isolated vertex then G □ P 2 and G □ C 2k have SCDCs. If G has an SCDC then so does G □ P k , k > 2 and G □ C 2k + 1 . We use these Cartesian results to show that P 2j [ G ] ( j ≥ 1) and C k [G] (k ≠ 3, 5, 7) have SCDCs. Also, if G has an SCDC then so does P 2j + 1 [G] (j ≥ 4). The results for the lexicographic product are harder and, in addition to the Cartesian results, require certain decompositions of K n,n into perfect matchings. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 99–123, 2008
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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 | 0.005 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 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 it