On the number of Hamiltonian cycles in the generalized Petersen graph
Bibliographic record
Abstract
The generalized Petersen graph <span class="math inline">\(G(n,k)\)</span> is a cubic graph with vertex set <span class="math inline">\(V(G(n,k))=\{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}\)</span> and edge set <span class="math inline">\(E(G(n,k))=\{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup \{v_i w_i\}_{0 \leq i < n}\)</span> where the indices are taken modulo <span class="math inline">\(n\)</span>. Schwenk found the number of Hamiltonian cycles in <span class="math inline">\(G(n,2)\)</span>, and in this article we present initial conditions and linear recurrence relations for the number of Hamiltonian cycles in <span class="math inline">\(G(n,3)\)</span> and <span class="math inline">\(G(n,4)\)</span>. This is attained by introducing <span class="math inline">\(G'(n,k)\)</span>, which is a modified version of <span class="math inline">\(G(n,k)\)</span>, and a subset of its subgraphs which we call admissible, and which are partitioned into different classes in such a manner that we can find relations between the number of admissible subgraphs of each class. The classes and their relations define a directed graph such that each strongly connected component is of a manageable size for <span class="math inline">\(k=3\)</span> and <span class="math inline">\(k=4\)</span>, which allows us to find linear recurrence relations for the number of admissible subgraphs in each class in these cases. The number of Hamiltonian cycles in <span class="math inline">\(G(n,k)\)</span> is a sum of the number of admissible subgraphs of <span class="math inline">\(G'(n,k)\)</span> over a certain subset of the classes.
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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 | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 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 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".