A Lower Bound for Radio k-chromatic Number of an Arbitrary Graph
Bibliographic record
Abstract
Radio $k$-coloring is a variation of Hale's channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph $G$, subject to certain constraints involving the distance between the vertices. Specifically, for any simple connected graph $G$ with diameter $d$ and a positive integer $k$, $1\leq k \leq d$, a radio $k$-coloring of $G$ is an assignment $f$ of positive integers to the vertices of $G$ such that $|f(u)-f(v)|\geq 1+k-d(u, v)$, where $u$ and $v$ are any two distinct vertices of $G$ and $d(u, v)$ is the distance between $u$ and $v$. In this paper we give a lower bound for the radio $k$-chromatic number of an arbitrary graph in terms of $k$, the total number of vertices $n$ and a positive integer $M$ such that $d(u,v)+d(v,w)+d(u,w)\leq M$ for all $u,v,w\in V(G)$. If $M$ is the triameter we get a better lower bound. We also find the triameter $M$ for several graphs, and show that the lower bound obtained for these graphs is sharp for the case $k=d$.
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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 | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.015 | 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".