A Note on Polynomials and $f$-Factors of Graphs
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Bibliographic record
Abstract
Let $G = (V,E)$ be a graph, and let $f : V \rightarrow 2^{\Bbb Z}$ be a function assigning to each $v \in V$ a set of integers in $\{0,1,2,\dots,d(v)\}$, where $d(v)$ denotes the degree of $v$ in $G$. Lovász defines an $f$-factor of $G$ to be a spanning subgraph $H$ of $G$ in which $d_{H}(v) \in f(v)$ for all $v \in V$. Using the combinatorial nullstellensatz of Alon, we prove that if $|f(v)| > \lceil {1\over 2}d(v) \rceil$ for all $v \in V$, then $G$ has an $f$-factor. This result is best possible and verifies a conjecture of Addario-Berry, Dalal, Reed and Thomason.
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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 | 0.001 | 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 it