Zonal and cozonal labelings using arbitrary abelian groups
Bibliographic record
Abstract
Let \(G\) be a plane graph with vertex, edge, and region sets \(V(G), E(G), F(G)\) respectively. A zonal labeling of a plane graph \(G\) is a labeling \(\ell: V(G)\rightarrow \{1,2\}\subset \mathbb{Z}_3\) such that for every region \(R\in F(G)\) with boundary \(B_R\), \(\sum\limits_{v\in V(B_R)}\ell(v)=0\) in \(\mathbb{Z}_3\). We extend this to general abelian groups, defining a \(\Gamma\)-zonal labeling as a labeling \(\ell:V(G)\rightarrow \Gamma\setminus \{0\}\) such that for every region \(R\in F(G)\), \(\sum\limits_{v \in V(B_R)}\ell(v)\) is \(0\). We explore existence of \(\Gamma\)-zonal labelings for various families of graphs. We also introduce two variations: generative and strong \(\Gamma\)-zonal labelings. A generative \(\Gamma\)-zonal labeling is one in which the elements used to label the vertices generate the group \(\Gamma\). A strong \(\Gamma\)-zonal labeling is a labeling in which the additive order of \(\ell(v)\) is equal to \(\deg(v).\) Examples and existence results are provided for both variations. It is shown that strong \(\Gamma\)-zonal labelings have a connection to edge colorings that generalizes the connection between zonal labelings and proper edge \(3\)-colorings of cubic maps.
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 machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.013 | 0.003 |
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 source (direct Gemma or distilled Codex), 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".