A neighborhood condition for graphs to have restricted fractional (g,f)-factors
Bibliographic record
Abstract
Let $h$ be a function defined on $E(G)$ with $h(e)\in[0,1]$ for any $e\in E(G)$. Set $d_G^{h}(x)=\sum_{e\ni x}h(e)$. If $g(x)\leq d_G^{h}(x)\leq f(x)$ for every $x\in V(G)$, then we call the graph $F_h$ with vertex set $V(G)$ and edge set $E_h$ a fractional $(g,f)$-factor of $G$ with indicator function $h$, where $E_h=\{e:e\in E(G),h(e)>0\}$. Let $M$ and $N$ be two sets of independent edges of $G$ with $M\cap N=\emptyset$, $|M|=m$ and $|N|=n$. If $G$ admits a fractional $(g,f)$-factor $F_h$ such that $h(e)=1$ for any $e\in M$ and $h(e)=0$ for any $e\in N$, then we say that $G$ has a fractional $(g,f)$-factor with the property $E(m,n)$. In this paper, we present a neighborhood condition for the existence of a fractional $(g,f)$-factor with the property $E(1,n)$ in a graph. Furthermore, it is shown that the neighborhood condition is sharp.
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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 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.003 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 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".