Bibliographic record
Abstract
For a set $\mathcal{S}$ of connected graphs, a subgraph $F$ of a graph $G$ is defined as an $\mathcal{S}$-factor of $G$ if $F$ satisfies that $V(F)=V(G)$ and every component of $F$ is isomorphic to an element of $\mathcal{S}$. If every component of $F$ is a star, then $F$ is said to be a star-factor. A star-factor with size at most $n$ may be written for a $\{K_{1,t}: 1\leq t\leq n\}$-factor. A graph $G$ is called a $\{K_{1,t}: 1\leq t\leq n\}$-factor deleted graph if $G-e$ has a $\{K_{1,t}: 1\leq t\leq n\}$-factor for every $e\in E(G)$. The sun toughness of a graph $G$ is denoted by $s(G)$ and defined as follows :$$ s(G)=\min \big\{\frac{|X|}{sun(G-X)}: X\subseteq V(G), \ sun(G-X)\geq2 \big\} $$ if $G$ is not a complete graph, and $s(G)=+\infty$ if $G$ is a complete graph, where $sun(G-X)$ denotes the number of sun components of $G-X$. In this paper, we prove that (1) if $G$ is a connected graph, and its sun toughness satisfies $s(G)\geq\frac{1}{n}$, then $G$ admits a $\{K_{1,t}: 1\leq t\leq n\}$-factor; (2) if $G$ is a $(k+1)$-connected graph, and its sun toughness $s(G)>\frac{k+1}{n+1}$, then $G-Y$ admits a $\{K_{1,t}: 1\leq t\leq n\}$-factor for any $Y\subseteq V(G)$ with $|Y|=k$; (3) if $G$ is a 2-edge-connected graph, and its sun toughness $s(G)\geq\frac{1}{n-1}$, then $G$ is a $\{K_{1,t}: 1\leq t\leq n\}$-factor deleted graph. Furthermore, it is shown that our results are 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 | 0.004 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.003 |
| Bibliometrics | 0.004 | 0.005 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.024 | 0.004 |
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".