\(e\)-Injective Coloring: \(2\)-Distance and Injective Coloring Conjectures
Bibliographic record
Abstract
An injective coloring of a given graph \(G = (V, E)\) is a vertex coloring of \(G\) such that any two vertices with a common neighbor receive distinct colors. An \(e\)-injective coloring of a graph \(G\) is a vertex coloring of \(G\) in which any two vertices \(v, u\) with a common edge \(e\) (\(e \neq uv\)) receive distinct colors; in other words, any two end vertices of a path \(P_4\) in \(G\) achieve different colors. With this new definition, we want to take a review of injective coloring of a graph from the new point of view. For this purpose, we review the conjectures raised so far in the literature of injective coloring and \(2\)-distance coloring, from the new approach of \(e\)-injective coloring. Additionally, we prove that, for disjoint graphs \(G, H\), with \(E(G) \neq \emptyset\) and \(E(H) \neq \emptyset\), \(\chi_{ei}(G \cup H) = \max\{\chi_{ei}(G), \chi_{ei}(H)\}\) and \(\chi_{ei}(G \vee H) = |V(G)| + |V(H)|.\) The \(e\)-injective chromatic number of \(G\) versus the maximum degree and packing number of \(G\) is investigated, and we denote \(\max\{\chi_{ei}(G), \chi_{ei}(H)\} \leq \chi_{ei}(G \square H) \leq \chi_{2}(G)\chi_{2}(H).\) Finally, we prove that, for any tree \(T\) (\(T\) is not a star), \(\chi_{ei}(T) = \chi(T),\) and we obtain the exact value of the \(e\)-injective chromatic number for some specified 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 | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".