Spanning<i>k</i>‐arc‐strong subdigraphs with few arcs in<i>k</i>‐arc‐strong tournaments
Bibliographic record
Abstract
Abstract Given a k ‐arc‐strong tournament T , we estimate the minimum number of arcs possible in a k ‐arc‐strong spanning subdigraph of T . We give a construction which shows that for each k ≥ 2, there are tournaments T on n vertices such that every k ‐arc‐strong spanning subdigraph of T contains at least $nk + {k(k-1)\over 2}$ arcs. In fact, the tournaments in our construction have the property that every spanning subdigraph with minimum in‐ and out‐degree at least k has $nk + {k(k-1)\over 2}$ arcs. This is best possible since it can be shown that every k ‐arc‐strong tournament contains a spanning subdigraph with minimum in‐ and out‐degree at least k and no more than $nk + {k(k-1)\over 2}$ arcs. As our main result we prove that every k ‐arc‐strong tournament contains a spanning k ‐arc‐strong subdigraph with no more than $nk + 136k^2$ arcs. We conjecture that for every k ‐arc‐strong tournament T , the minimum number of arcs in a k ‐arc‐strong spanning subdigraph of T is equal to the minimum number of arcs in a spanning subdigraph of T with the property that every vertex has in‐ and out‐degree at least k . We also discuss the implications of our results on related problems and conjectures. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 265–284, 2004
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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 | 0.004 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.003 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| 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".