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 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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| 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".