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 2 c + 1. If G is a forest, then there is a tree T whose G ‐spectrum contains both 2 and 2 c . 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 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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".