Bibliographic record
Abstract
If $G$ is a $k$-chromatic graph of order $n$ then it is known that the chromatic polynomial of $G$, $π(G,x)$, is at most $x(x-1)\cdots (x-(k-1))x^{n-k} = (x)_{\downarrow k}x^{n-k}$ for every $x\in \mathbb{N}$. We improve here this bound by showing that \[ π(G,x) \leq (x)_{\downarrow k} (x-1)^{Δ(G)-k+1} x^{n-1-Δ(G)}\] for every $x\in \mathbb{N},$ where $Δ(G)$ is the maximum degree of $G$. Secondly, we show that if $G$ is a connected $k$-chromatic graph of order $n$ where $k\geq 4$ then $π(G,x)$ is at most $(x)_{\downarrow k}(x-1)^{n-k}$ for every real $x\geq n-2+\left( {n \choose 2} -{k \choose 2}-n+k \right)^2$ (it had been previously conjectured that this inequality holds for all $x \geq k$). Finally, we provide an upper bound on the moduli of the chromatic roots that is an improvment over known bounds for dense graphs.
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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 | 0.004 | 0.028 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.006 | 0.006 |
| Science and technology studies | 0.004 | 0.008 |
| Scholarly communication | 0.006 | 0.021 |
| Open science | 0.006 | 0.009 |
| Research integrity | 0.002 | 0.017 |
| Insufficient payload (model declined to judge) | 0.031 | 0.007 |
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".