A Problem on Edge-magic Labelings of Cycles
Bibliographic record
Abstract
Abstract In 1970, Kotzig and Rosa defined the concept of edge-magic labelings as follows. Let G be a simple (p; q)-graph (that is, a graph of order p and size q without loops or multiple edges). A bijective function f : V(G)⊔E(G) → {1; 2, … p+q} is an edge-magic labeling of G if f (u)+ f (uv)+ f (v) = k, for all uv ∊ E(G). A graph that admits an edge-magic labeling is called an edge-magic graph, and k is called the magic sum of the labeling. An old conjecture of Godbold and Slater states that all possible theoretical magic sums are attained for each cycle of order n ≥ 7. Motivated by this conjecture, we prove that for all n0∊ N, there exists n 2 N such that the cycle Cn admits at least n0 edge-magic labelings with at least n0 mutually distinct magic sums. We do this by providing a lower bound for the number of magic sums of the cycle Cn, depending on the sum of the exponents of the odd primes appearing in the prime factorization of n.
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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 | 0.002 | 0.011 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.003 | 0.003 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
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".