Toughness of $K_{a,t}$-Minor-free Graphs
Bibliographic record
Abstract
The toughness of a non-complete graph $G$ is the minimum value of $\frac{|S|}{\omega(G-S)}$ among all separating vertex sets $S\subset V(G)$, where $\omega(G-S)\ge 2$ is the number of components of $G-S$. It is well-known that every $3$-connected planar graph has toughness greater than $1/2$. Related to this property, every $3$-connected planar graph has many good substructures, such as a spanning tree with maximum degree three, a $2$-walk, etc. Realizing that 3-connected planar graphs are essentially the same as 3-connected $K_{3,3}$-minor-free graphs, we consider a generalization to $a$-connected $K_{a,t}$-minor-free graphs, where $3\le a\le t$. We prove that there exists a positive constant $h(a,t)$ such that every $a$-connected $K_{a,t}$-minor-free graph $G$ has toughness at least $h(a,t)$. For the case where $a=3$ and $t$ is odd, we obtain the best possible value for $h(3,t)$. As a corollary it is proved that every such graph of order $n$ contains a cycle of length $\Omega(\log_{h(a,t)} n)$.
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How this classification was reachedexpand
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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.005 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".