On (k, t)-choosability of graphs.
Bibliographic record
Abstract
A k-list assignment L of a graph G is a mapping which assigns to each vertex v of G a set L(v) of size k. A (k,t)-list assignment of G is a k-list assignment with | ⋃ v∈V (G) L(v)| = t. An L-coloring φ of G is a proper coloring of G such that φ(v) is chosen from L(v) for every vertex v. A graph G is Lcolorable if G has an L-coloring. When the parameter t is not of special interest, we simply say k-list assignment. Particularly, if L is a (k, k)-list assignment of G, then any L-coloring is called a k-coloring for G. A graph G is (k, t)-choosable if G is L-colorable for every (k, t)-list assignment L. If a graph G is (k, t)-choosable for any number t then G is k-choosable and the smallest number k satisfying this properties is called the list chromatic number of G denoted by χl(G). The list coloring problem is first studied by Vizing[6] and by Erdos, Rubin and Taylor[2]. In [2], the authors give a characterization of 2-choosable graphs. There is no literature giving a characterization of k-choosable graphs for k ≥ 3. The k-choosability of graphs is revealed only for some specific classes of graphs. For example, Thomassen[5] proves that every planar graph is 5-choosable while some planar graphs are 3-choosable. (See [4],[8],[7],[3],[9] and [10].) Ganjari et al. [1] use (k, t)-choosability of graphs to characterize uniquely 2-list colorable graphs. When k ≥ χl(G), a graph G is always (k, t)-choosable. In this paper, we focus on any integer k such that k < χl(G). For an n-vertex graph G, we find the value t in terms of n and k such that G is (k, t)-choosable. Our main study includes the following results. For fixed numbers n, k and t, every n-vertex graph is (k, t)-choosable if and only if t ≥ kn − k2 + 1. In case k ≤ t ≤ kn − k2, every n-vertex graph containing Kk+1 is not (k, t)choosable. Furthermore, every Kk+1-free n-vertex graph is (k, t)-choosable if and only if t ≥ kn−k2−2k. If k ≤ t ≤ kn−k2−2k+1, an n-vertex graph
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.005 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.003 | 0.011 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.013 | 0.002 |
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".