Graph Products with Small Cycle Double Covers
Bibliographic record
Abstract
A cycle double cover of a graph G is a collection of cycles, C, such that every edge of G lies in precisely two cycles of C. The Small Cycle Double Cover (SCDC) Conjecture, proposed by J.A. Bondy, asserts that every simple bridgeless graph on n vertices has a cycle double cover with at most n-1 cycles, and is a strengthening of the well-known Cycle Double Cover (CDC) Conjecture. Both the CDC Conjecture and the SCDC Conjecture have been verified for various classes of graphs, but remain open in general. The graphs for which that SCDC Conjecture has been verified all have well defined structural properties that play an important role. The structure that is inherent in graph products makes such graphs ideal special cases for which to verify the SCDC Conjecture. There are various graph products that can be considered, and in this talk I will describe some results and techniques for proving the SCDC Conjecture for certain graph products. This talk will be accessible to a general mathematics audience: all relevant terms will be defined, and proofs will be illustrated with examples. This is joint work with R.J. Nowakowski (Dalhousie University).
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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 | 0.000 | 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 | 0.000 |
| Insufficient payload (model declined to judge) | 0.072 | 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".