The <i>f</i>-Chromatic Index of a Graph Whose <i>f</i>-Core Has Maximum Degree 2
Bibliographic record
Abstract
Abstract. Let G be a graph. The minimum number of colors needed to color the edges of G is called the chromatic index of G and is denoted by χ'(G). It is well known that , for any graph G, where Δ(G) denotes the maximum degree of G. A graph G is said to be class 1 if x'(G) = Δ(G) and class 2 if χ'(G) = Δ(G)+1. Also, GΔ is the induced subgraph on all vertices of degree Δ(G). Let f : V(G) → ℕ be a function. An f-coloring of a graph G is a coloring of the edges of E(G) such that each color appears at each vertex v ∊ V(G) at most f (v) times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by χ'f (G). It was shown that for every graph , where . A graph G is said to be f -class 1 , and f -class 2, otherwise. Also, GΔf is the induced subgraph of G on . Hilton and Zhao showed that if G has maximum degree two and G is class 2, then G is critical, GΔ is a disjoint union of cycles and δ(G) = Δ(G)–1, where δ(G) denotes the minimum degree of G, respectively. In this paper, we generalize this theorem to f -coloring of graphs. Also, we determine the f -chromatic index of a connected graph G with |GΔf| ≤ 4.
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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 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 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".