Binding Number, Toughness and General Matching Extendability in Graphs
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
A connected graph $G$ with at least $2m + 2n + 2$ vertices which contains a perfect matching is $E(m, n)$-{\it extendable}, if for any two sets of disjoint independent edges $M$ and $N$ with $|M| = m$ and $|N|= n$, there is a perfect matching $F$ in $G$ such that $M\subseteq F$ and $N\cap F=\emptyset$. Similarly, a connected graph with at least $n+2k+2$ vertices is called $(n,k)$-{\it extendable} if for any vertex set $S$ of size $n$ and any matching $M$ of size $k$ of $G-S$, $G-S-V(M)$ contains a perfect matching. Let $\varepsilon$ be a small positive constant, $b(G)$ and $t(G)$ be the binding number and toughness of a graph $G$. The two main theorems of this paper are: for every graph $G$ with sufficiently large order, 1) if $b(G)\geq 4/3+\varepsilon$, then $G$ is $E(m,n)$-extendable and also $(n,k)$-extendable; 2) if $t(G)\geq 1+\varepsilon$ and $G$ has a high connectivity, then $G$ is $E(m,n)$-extendable and also $(n,k)$-extendable. It is worth to point out that the binding number and toughness conditions for the existence of the general matching extension properties are almost same as that for the existence of perfect matchings. Comment: 10 pages
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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 | 0.006 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.004 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.006 | 0.015 |
| Research integrity | 0.000 | 0.002 |
| 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