\(\left(s,s+1,s+1,\dots\right)\)-packing colorings and \(\left(1,s,s,\dots\right)\)-packing colorings and d-distance colorings of distance graphs \(D(1,t)\)
Bibliographic record
Abstract
An S-packing k-coloring of a graph \(G\) (with \(S=(s_1,s_2,\dots)\) is a non-decreasing sequence of positive integers) is a mapping \(f\) from \(V(G)\) to \(\lbrace 1,\dots , k \rbrace\) (the set of colors) such that for every two distincts vertices \(x\) and \(y\) in \(V(G)\) with \(f(x)=f(y)=i\) the distance between \(x\) and \(y\) in \(G\) is bigger than \(s_i\). The S-packing chromatic number \(\chi_S (G)\) of \(G\) is the smallest integer \(k\) such that \(G\) has an S-packing k-coloring. Given a set \(D\subset \mathbb{N}^*\), a distance graph \(G(\mathbb{Z}, D)\) with distance set \(D\) is a graph with vertex set \(\mathbb{Z}\) and two distincts vertices \(u\) and \(v\) are adjacents if \(| u-v | \in D\). In this paper, for \(S=(s,s+1,s+1,\dots)\) with \(s \geq \left\lceil \frac{t}{2} \right\rceil\) we give a lower bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\), and a lower bound of \(\chi_d (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) with \(d \geq \left\lceil \frac{t}{2} \right\rceil\), for \(S=(s_1,s_2,\dots, s_i,a,a,\dots)\) with \(a \geq \max( 1 , t-2 )\) we give an upper bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\), and we determine the exact values of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) and also of \(\chi_d (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) for \(s\geq \max (\left\lceil \frac{t}{2} \right\rceil, t-3)\) and \(d \geq \max (\left\lceil \frac{t}{2} \right\rceil , t-2)\). And we give a lower and an upper bound of \(\chi_S (G(\mathbb{Z}, \lbrace 1, t\rbrace))\) for \(S=(1,s,s,\dots)\) with conditions on \(s\) and \(t\), which in the cases \(s\geq \max (t-2,\left\lceil \frac{t}{2}\right\rceil)\) we determine the exact values of \(\chi_S (G(\mathb b{Z}, \lbrace 1, t\rbrace))\).
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.038 | 0.009 |
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".