Circulant Distant Two Labeling and Circular Chromatic Number
Bibliographic record
Abstract
Let G be a graph and d,d 0 be positive integers, d 0 d. An m-(d,d 0 )-circular distance two labeling is a function f : V (G) ! {0,1,2,···,m 1} such that |f(u) f(v)|m d if u and v are adjacent; and |f(u) f(v)|m d 0 if u and v are distance two apart, where |x|m := min{|x|,m| x|}. The minimum m such that there exists an m-(d,d 0 )-circular labeling for G is called the d,d0-number of G and denoted by d,d 0(G). The d,d 0-numbers for trees can be obtained by a first-fit algorithm. In this article, we completely determine the d,1-numbers for cycles. In addition, we show connections between generalized circular distance labeling and circular chromatic number.
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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 | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".