First-Fit coloring of Cartesian product graphs and its defining sets
Bibliographic record
Abstract
Let the vertices of a Cartesian product graph G□H be ordered by an ordering σ. By the First-Fit coloring of (G□H,σ) we mean the vertex coloring procedure which scans the vertices according to the ordering σ and for each vertex assigns the smallest available color. Let FF(G□H,σ) be the number of colors used in this coloring. By introducing the concept of descent we obtain a sufficient condition to determine whether FF(G□H,σ)=FF(G□H,τ), where σ and τ are arbitrary orders. We study and obtain some bounds for FF(G□H,σ), where σ is any quasi-lexicographic ordering. The First-Fit coloring of (G□H,σ) does not always yield an optimum coloring. A greedy defining set of (G□H,σ) is a subset S of vertices in the graph together with a suitable pre-coloring of S such that by fixing the colors of S the First-Fit coloring of (G□H,σ) yields an optimum coloring. We show that the First-Fit coloring and greedy defining sets of G□H with respect to any quasi-lexicographic ordering (including the known lexicographic order) are all the same. We obtain upper and lower bounds for the smallest cardinality of a greedy defining set in G□H, including some extremal results for Latin squares.
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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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".