On the chromatic spectrum of acyclic decompositions of graphs
Bibliographic record
Abstract
Abstract If G is any graph, a G‐decomposition of a host graph H = (V, E) is a partition of the edge set of H into subgraphs of H which are isomorphic to G. The chromatic index of a G‐decomposition is the minimum number of colors required to color the parts of the decomposition so that two parts which share a node get different colors. The G‐spectrum of H is the set of all chromatic indices taken on by G‐decompositions of H. If both S and T are trees, then the S‐spectrum of T consists of a single value which can be computed in polynomial time. On the other hand, for any fixed tree S, not a single edge, there is a unicyclic host whose S‐spectrum has two values, and if the host is allowed to be arbitrary, the S‐spectrum can take on arbitrarily many values. Moreover, deciding if an integer k is in the S‐spectrum of a general bipartite graph is NP‐hard. We show that if G has c > 1 components, then there is a host H whose G‐spectrum contains both 3 and 2c + 1. If G is a forest, then there is a tree T whose G‐spectrum contains both 2 and 2c. Furthermore, we determine the complete spectra of both paths and cycles with respect to matchings. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 83–104, 2007
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".