Bibliographic record
Abstract
A mapping l:E(G)→A, where A is an abelian group written additively, is called an edge labeling of the graph G. For every positive integer h⩾2, a graph G is said to be zero-sum h-magic if there is an edge labeling l from E(G) to Zh∖{0} such that s(v)=∑uv∈E(G)l(uv)=0 for every vertex v∈V(G). In 2014, Saieed Akbari, Farhad Rahmati and Sanaz Zare proved that if r (r≠5) is odd and G is a 2-edge connected r-regular graph, G admits a zero-sum 3-magic labeling, and they also conjectured that every 5-regular graph admits a zero-sum 3-magic. In this paper, we first prove that every 5-regular graph with every edge contained in a triangle must have a perfect matching, and then we denote the edge set of the perfect maching by EM, and we make a labeling l:E(EM)→2, and E(E(G)–EM)→1. Thus we can easily see this labeling is a zero-sum 3-magic, confirming the above conjecture with a moderate condition.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".