Bibliographic record
Abstract
An additive labeling of a graph $G$ is a function $\ell :V(G) \rightarrow \mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\Sigma_{w \sim v} \ell (w) \neq \Sigma_{w \sim u} \ell (w)$ ($x \sim y$ means that $x$ is joined to $y$). The additive number of $G$, denoted by $\eta (G)$, is the minimum number $k$ such that $G$ has a additive labeling $\ell : V(G) \rightarrow \mathbb{N}_k$. The additive choosability of a graph $G$, denoted by $\eta_\ell (G)$, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list. Seamone in his PhD thesis conjectured that for every graph $G$, $\eta(G)= \eta_\ell (G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $\eta_\ell (G) - \eta(G) \geq k$. A $(0,1)$-additive labeling of a graph $G$ is a function $\ell :V(G) \rightarrow \{0,1 \}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\Sigma_{w \sim v} \ell (w) \neq \Sigma_{w \sim u} \ell (w)$. A graph may lack any $(0,1)$-additive labeling. We show that it is NP-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $\sigma_1 (G) = \mathrm{min}_{\ell \in \Gamma} \Sigma_{v \in V (G)} \ell (v)$ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $\epsilon > 0$ , approximating the $(0,1)$-additive number within $n^{1-\epsilon}$ is NP-hard.
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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 | 0.022 |
| Meta-epidemiology (narrow) | 0.004 | 0.002 |
| Meta-epidemiology (broad) | 0.004 | 0.005 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.003 | 0.008 |
| Scholarly communication | 0.010 | 0.015 |
| Open science | 0.006 | 0.005 |
| Research integrity | 0.005 | 0.012 |
| Insufficient payload (model declined to judge) | 0.016 | 0.005 |
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".