Odd‐angulated graphs and cancelling factors in box products
Bibliographic record
Abstract
Abstract Under what conditions is it true that if there is a graph homomorphism G □ H → G □ T , then there is a graph homomorphism H → T ? Let G be a connected graph of odd girth 2 k + 1. We say that G is (2 k + 1)‐angulated if every two vertices of G are joined by a path each of whose edges lies on some (2 k + 1)‐cycle. We call G strongly (2 k + 1)‐angulated if every two vertices are connected by a sequence of (2 k + 1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2 k + 1)‐angulated, H is any graph, S , T are graphs with odd girth at least 2 k + 1, and ϕ: G □ H → S □ T is a graph homomorphism, then either ϕ maps G □{ h } to S □{ t h } for all h ∈ V ( H ) where t h ∈ V ( T ) depends on h ; or ϕ maps G □{ h } to { s h }□ T for all h ∈ V ( H ) where s h ∈ V ( S ) depends on h . This theorem allows us to prove several sufficient conditions for a cancelation law of a graph homomorphism between two box products with a common factor. We conclude the article with some open questions. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:221‐238, 2008
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 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".