Bibliographic record
Abstract
An edge-coloring of a graph \(G\) with natural numbers \(1,2,\ldots\) is called a sum edge-coloring if the colors of edges incident to any vertex of \(G\) are distinct and the sum of the colors of the edges of \(G\) is minimum. The edge-chromatic sum of a graph \(G\) is the sum of the colors of edges in a sum edge-coloring of \(G\). In general, the problem of finding the edge-chromatic sum of an \(r\)-regular (\(r\geq 3\)) graph is \(NP\)-complete. In this paper we provide some bounds on the edge-chromatic sums of various products of graphs. In particular, we give tight upper bounds on the edge-chromatic sums of tensor, strong tensor, Cartesian, strong products and composition of graphs. We also determine the edge-chromatic sums and edge-strengths of the Cartesian products of regular graphs and paths (cycles) with an even number of vertices. Finally, we determine the edge-chromatic sums and edge-strengths of grids, cylinders, and tori.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.013 | 0.002 |
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".