Polychromatic colorings of certain subgraphs of complete graphs and maximum densities of substructures of a hypercube
Bibliographic record
Abstract
If G is a graph and H is a set of subgraphs of G, an edge-coloring of G is H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, polyHG, is the largest number of colors in an H-polychromatic coloring. We determine polyHG exactly when G is a complete graph on n vertices, q a fixed nonnegative integer, and H is the family of one of: all matchings spanning n-q vertices, all 2-regular graphs spanning at least n-q vertices, or all cycles of length precisely n-q. For H, K, subsets of the vertex set V(Qd) of the d-cube Qd, K is an exact copy of H if there is an automorphism of Qd sending H to K. For a positive integer, d, and a configuration in Qd, H, we define λ(H,d) as the limit as n goes to infinity of the maximum fraction, over all subsets S of V(Qn), of sub-d-cubes of Qn whose intersection with S is an exact copy of H. We determine λ(C8,4) and λ(P4,3) where C8 is a “perfect” 8-cycle in Q4 and P4 is a “perfect” path with 4 vertices in Q3, λ(H,d) for several configurations in Q2, Q3, and Q4, and an infinite family of configurations. Strong connections exist with extensions Ramsey numbers for cycles in a graph, counting sequences with certain properties, inducibility of graphs, and we determine the inducibility of two vertex disjoint edges in the family of bipartite graphs.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".