Expanders Are Universal for the Class of All Spanning Trees
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Bibliographic record
Abstract
A graph is called universal for a given graph class (or, equivalently, - universal ) if it contains a copy of every graph in as a subgraph. The construction of sparse universal graphs for various classes has received a considerable amount of attention. There is particular interest in tight -universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from . Arguably, the most studied case is that when is some class of trees. In this work, we are interested in ( n ,Δ), the class of all n -vertex trees with maximum degree at most Δ. We show that every n -vertex graph satisfying certain natural expansion properties is ( n ,Δ)-universal. Our methods also apply to the case when Δ is some function of n . Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constant c such that the random graph G(n,cn −1/3 log 2 n ) is asymptotically almost surely (a.a.s.) universal for ( n,O (1)). Moreover, a corresponding result holds for the random regular graph of degree cn 2/3 log 2 n . Another interesting consequence is the existence of locally sparse n -vertex ( n ,Δ)-universal graphs. For example, we show that one can (randomly) construct n -vertex ( n,O (1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose ( n ,Δ)-universal graphs with merely O(n) edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly ( n ,Δ)-universal in the context of the Maker–Breaker tree-universality game.
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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 | 0.001 |
| 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 | 0.001 | 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