On the number of longest and almost longest cycles in cubic graphs
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Bibliographic record
Abstract
We consider the questions: How many longest cycles must a cubic graph have, and how many may it have? For each k >= 2 there are infinitely many p such that there is a cubic graph with p vertices and precisely one longest cycle of length p-k. On the other hand, if G is a graph with p vertices, all of which have odd degree, and its longest cycle has length p-1, then it has a second (but not necessarily a third) longest cycle. We presents results and conjectures on the maximum number of cycles in cubic multigraphs of girth 2, 3, 4, respectively. For cubic cyclically 5-edge-connected graphs we have no conjecture but, we believe that the generalized Petersen graphs P(n, k) are relevant. We enumerate the hamiltonian and almost hamiltonian cycles in each P(n, 2). Curiously, there are many of one type if and only there are few of the other. If n is odd, then P(2n, 2) is a covering graph of P(n, 2). (For example, the dodecahedron graph is a covering graph of the Petersen graph). Another curiosity is that one of these has many (respectively few) hamiltonian cycles if and only if the other has few (respectively many) almost hamiltonian cycles.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 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 it