Couterexample to a conjecture on the structure of bipartite partionable graphs
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Bibliographic record
Abstract
A graph G is called a prism fixer if (G ×K2) = (G), where (G) denotes the domination number of G. A symmetric -set of G is a minimum dominating set D which admits a partition D = D1 ∪ D2 such that V (G) − N[Di] = Dj, i,j = 1,2, i 6 j. It is known that G is a prism fixer if and only if G has a symmetric -set. Hartnell and Rall [On dominating the Cartesian product of a graph and K2, Discuss. Math. Graph Theory 24 (2004), 389–402] conjectured that if G is a connected, bipartite graph such that V (G) can be partitioned into symmetric -sets, then G ∼ C4 or G can be obtained from K2t,2t by removing the edges of t vertex-disjoint 4-cycles. We construct a counterexample to this conjecture and prove an alternative result on the structure of such bipartite graphs.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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